Università di Pisa
Scuola di Ingegneria
Corso di laurea in Ingegneria Nucleare
Dipartimento di Ingegneria Civile e Industriale
CFD study of polonium evaporation
from LBE liquid mixture
Relatore:
Candidato:
Prof. Nicola Forgione
Fulvio Bertocchi
Correlatori:
Alessandro Marino
Borja Gonzalez
Of these sciences the gate and the key is mathematics. He who is ignorant of mathematics cannot know the other sciences nor the aairs of the world, as I shall prove. And what is worse men ignorant of this do not perceive their own ignorance and therefore do not seek a remedy. Roger Bacon, The Opus Majus
Contents
Abstract 1
1 Introduction 3
1.1 Energy demand . . . 3
1.2 Future challenges in the nuclear industry . . . 4
1.3 ADS: a possible solution for the nuclear waste . . . 6
1.4 Lead Bismuth Eutectic as coolant and spallation target . . . . 10
2 MYRRHA 11 2.1 Design . . . 12
2.2 Proton accelerator . . . 14
2.3 Spallation loop . . . 18
2.3.1 Window spallation target . . . 18
2.3.2 Windowless spallation target . . . 18
3 Polonium issue in MYRRHA 21 3.1 Evaporation of dissolved components . . . 22
3.2 Transpiration method . . . 25
3.2.1 Fundamentals . . . 26
3.2.2 Po diusion in LBE . . . 28
3.2.3 Evaporation . . . 31
3.2.4 Diusion of Po in carrier gas . . . 32
3.3 Po evaporation from LBE . . . 34
4 Mathematical models 38 4.1 Governing equations . . . 38
4.2 Turbulence models . . . 40
4.2.1 Boussinesq hypothesis . . . 40
4.2.2 Stress Transport models . . . 43
4.3 Mass transfer theory . . . 44
5 Obtained results 48 5.1 Preliminary remarks . . . 48
5.1.1 Unit of measure of the Henry constant . . . 48
5.1.2 Implementation of the integral function . . . 49
5.2 Modelling of the mass transfer inside the tube . . . 50
5.3 Estimation of the mass transfer coe cient . . . 54
5.4 Polonium release from LBE . . . 56
5.4.1 Sample geometry . . . 56
5.4.2 Diusion of Po in LBE . . . 58
5.4.3 Polonium depletion in LBE . . . 59
5.5 Mass transport inside the tube . . . 61
6 Multicomponent multiphase simulation 73 6.1 Governing equations . . . 73
6.2 Initialization . . . 74
6.3 Issues with con nement of LBE . . . 75
7 Conclusions 79 7.1 Summary . . . 79
7.2 Future works . . . 81
Appendices 82
A Mass transfer coe cient from Graetz-Nusselt approach 83
B Mass transfer from a at plate 86
Abstract
The need to nd a suitable solution for the disposal of radioactive waste has led to a growing interest in the last few decades in the development of accelerator-driven systems (ADS) which could e ciently perform the trans-mutation of this waste transforming it to shorter-lived, less hazardous ra-dioactive species.
One of the most prominent ADS projects worldwide is MYRRHA which consists of a subcritical core coupled to a proton accelerator. Lead-bismuth eutectic (LBE) is used both as coolant and spallation target, supported by ad-vantageous physicochemical properties. However it also presents some chal-lenges. One concern with the use of LBE as coolant is the formation of large quantities of 210-Po due to the neutron capture reaction with bismuth (Bi). Polonium is critical to safety of ADS due to its elevated radiotoxicity. A qualitative and quantitative understanding of Po evaporation from LBE is thus required for safety assessments, design and licensing of LBE-based nu-clear systems. With this aim Po evaporation experiments were performed at SCK•CEN resorting to the so-called transpiration method.
The setup is composed by a quartz tube containing Po-doped LBE sample (melted by an oven); Argon gas ows inside the tube and carries away the polonium vapours that evaporate outside the sample. The amount of evapo-rated polonium can be estimated with the collection of the Po-Ar gas mixture at the outlet of the tube.
The present work is focused on the Computational Fluid Dynamic (CFD) simulation of the transpiration experiments using ANSYS-CFX code. The aim was to investigate the mass transfer phenomena at the base of the ex-perimental tests and, if possible, to reproduce them numerically.
The dependency of the mass transfer coe cient of polonium in the gas phase on the gas velocity was determined by CFD calculations. The source term in these calculations requires an adequate choice of the boundary condition at the interface with the sample.
Secondly the Po-doped LBE sample was investigated. Polonium depletion in the LBE sample during the evaporation was studied aiming to evaluate the in uence that the polonium diusion has on its subsequent evaporation, and the variation of the polonium concentration pro le with the carrier gas ow
rate, close to the gas-liquid interface, inside the LBE sample. The calcula-tions revealed the independence of the amount of the evaporated material on the diusion coe cient of polonium in LBE.
Finally simulations of polonium diusion in the gas phase and its transport outside the tube were conducted. They were performed using the polonium concentration pro les close to the liquid-gas interface, calculated in the previ-ous part of the work inside the LBE sample, as boundary condition describing the Po release in the gas carrier.
The CFD analysis of the evaporation and convective transport of polonium inside the tube allowed to get results that may be useful as validation of the hypothesis considered in the experiments.
Moreover, a parametric study on the turbulence models was conducted. The obtained results emphasized the importance of properly simulating the dif-fusion boundary layer above the sample as it is an in uencing step on the overall phenomenon of evaporation and convection of polonium inside the tube.
1
Introduction
1.1 Energy demand
With the current growth in the electricity demand, an alternative energy source to the commonly used fossil fuels is required to decrease the CO2 emissions which are the main contributor to the global warming, so-called greenhouse eect.
These terms were already used in 1863 by the scientist J.Tyndall and ca.30 years later S.Arrhenius attempted the rst evaluation of the impact of a CO2 increase in the atmosphere. Nowadays the greenhouse eect is a
well-known phenomenon undermining the critical equilibrium of production and absorption of carbon dioxide: over 28 billion tonnes of CO2 are emitted each
year in the environment by human activities. Even though this amount is only 3% of the CO2 coming from natural sources, it is a serious concern as
CO2 gives raise to 60% of the increase of the greenhouse eect originated
from human activities.
Besides the greenhouse eect, the increase in the oil price at the beginning of the 1970's led to the search of alternative and more e cient energy sources. These alternative energy sources should account for an increase by 50% in the energy demand is expected within 2030 due to the growth on the overall population worldwide (expected to be ca.8 billion within the year 2025) [1] (Figure 1.1).
Figure 1.1: Energy demand over the last 60 years. [2]
Based on the most reliable predictions, the population is expected to reach 8 billion within the year 2025, leading to double the amount of required electricity by 2030. To satisfy this growth in the energy demand, besides the typical nite energy sources (fossil, fuels..), alternative options which also meet the requirements of lower CO2 emissions, are increasing their share in the global energy production.
1.2 Future challenges in the nuclear industry
Nuclear energy shows many of the typical aspects associated with the so called renewable sources, as low carbon emissions and small environmental impact. However nuclear energy cannot be classi ed as a pure renewable source as the most commonly used fuel in typical ssion reactors, uranium, is nite. Moreover only a small fraction of 235U present in the mined ore is
employed as ssile material, though it undergoes an enrichment process to increase up to 3 − 5% its presence on the total amount of uranium fuel of a reactor.The most common type of nuclear reactors is based on the ssion of 235U under a thermal neutron ux. 235U, the only natural ssile isotope
under these neutron uxes, represents only 0.7% of the naturally occurring uranium.
The ine ciency in the use of the uranium resources leads to the generation of large amount of nuclear waste. The disposal of these products is one of the major issues of the nuclear community nowadays. The waste is classi ed according to its radioactivity levels in a range that spans from the LLW (low level radioactive waste) to the HLW (high level radioactive waste). The latter class is the more relevant due to the very long time required to decrease its activity caused by minor actinides resulting from the ssion reaction, such as americium, neptunium and curium.
suitable disposal of the current nuclear waste, is thus required.
To overcome the problem of the low e ciency in the use of the uranium sources the next generation of nuclear power plants (generation IV) will make use of fast neutrons which allows the ssion of not only235U but also of 238U
with the consequent continuous generation of239P u.
Nowadays 1.5 million tonnes of depleted uranium are a potential nuclear waste because of the dominance of U 238. The future generation will improve the e ciency of its use getting close to the closure of the fuel cycle.
Figure 1.2: Fission cross sections of ssile and fertile materials. [3] The ssion cross section of fertile elements such as Thorium or 238U is
almost zero at energies typical of thermal neutrons uxes used in ssion reac-tors. These ssion cross sections however increase at higher neutron energies (Figure 1.2). The process of conversion from fertile to ssile materials is called breeding, hence the name of fast breeder reactors.
The breeding reactions are listed below:
238 92 U +10n γ −−−−−−−−→239 92 U β− −−−−−−−−→239 93 N p β− −−−−−−−−→239 94 P u (1.1) 232 90 T h + 1 0n γ −−−−−−−−→233 90 T h β− −−−−−−−−→233 91 P a β− −−−−−−−−→233 92 U (1.2)
Even though 20 nuclear fast reactors have already been operated since 1950, the search of a solution for the afore mentioned problems has led to an growing interest on this sort of fast neutron systems. For instance, countries such as France, who scheduled to replace half of its nuclear capacity by fast reactors by 2050 and which is working on ASTRID with sodium as coolant, and the JAEA (Japan Atomic Energy Agency) that is studying the project
of a reactor to substitute the FBR Monju are heading to this sort of nuclear facilities, among which the main options are Sodium-Cooled Fast Reactor (SFR), Gas-Cooled Fast Reactor (GCFR), Supercritical Water-cooled Fast Reactor (SCWR), Very High Temperature Reactor (VHTR), Molten Salt Reactor (MSR) and Liquid Metal Fast Reactor (LMFR).
1.3 ADS: a possible solution for the nuclear
waste
The main problem with the fast spectrum generation IV nuclear reactors is the load capability of transuranium elements such as 239P u and minor
ac-tinides: when this fraction increases, the reactivity of the reactor may be uncontrolled due to the small quantity of delayed neutrons available to con-trol the reactor. The accelerator driven system (ADS) was proposed as a possible solution to overcome reactivity issue.
It consists on coupling of a proton accelerator with a spallation target and a subcritical core, which leads to a more e cient transmutation of minor actinides in lighter elements reducing reactivity problems.
Besides the larger loads of minor actinides ADS also oers an intrinsic safety feature related with the sub critical core as an external neutron source is needed to sustain the nuclear chain reaction. This neutron source results from the collision of the proton beam with the spallation target. Therefore, in case of severe emergency, as loss of on site or o site electric supply, the system could be shut down just by stopping the proton beam.
The spallation reaction
The spallation target is the medium where the spallation reaction that multi-plies neutrons takes place: it starts when an energetic particle hits a nucleus, leading to a production of secondary particle swarm (Figure 1.3).
Figure 1.3: Scheme of spallation reaction. [4]
In ADS, the spallation target is one of the key elements owing to the need of generating neutrons that sustain the ssion reaction chains in the sub-critical core. There are dierent routes for the production of these neu-trons. Among these routes, one consists on the bombardment of a spallation target by a proton beam so that a series of spallation reactions is triggered. This source goes under the name of spallation target" due to the spallation reaction that takes place therein.
A general de nition of spallation reaction does not exist even though it is a process observed in a variety of elds as the astronomy, radiotherapy or geophysics; in practical terms there is a non elastic interaction between en-ergetic particles like protons, pions or neutrons and an atomic nucleus. In Encyclopedia Britannica spallation is de ned as an high-energy nuclear re-action in which a target nucleus struck by an incident(bombarding) particle, of energy greater than 50 MeV, ejects numerous lighter particles and be-comes a product nucleus correspondingly lighter than the original nucleus. The light ejected particle may be neutrons, protons, or various composite particles equivalent" [4].
The rst attempt to explain the process was conducted by Robert Serber who proposed that, at energies around 100 MeV, the deBroglie wavelength of the bombarding particle becomes of the same order of magnitude, of the inter-nucleon distance and then able to interact with the individual nucleons.
λ = h p2mpEp
When the target is thick enough to permit also secondary reactions, a "nu-clear cascade" will occur, like the one induced by the incoming cosmic rays in the atmosphere. The rst who studied this radiation was the Nobel prize winner Hess in 1912 by means of free balloon ights and he named it H ¨ohenstralung (high altitude rays); the primary are constituted by protons and alpha particles while, after interactions with the atmosphere's molecules, the secondary rays are formed and they are composed by neutrinos,gamma rays and pions decaying in muons.
The initial interaction between the spallation target and the incoming beam of light particles with a kinetic energy of several hundred of MeV provokes the ejection of small groups of nucleons from the nucleus leaving it in a excited state. This stage is the so-called intra-nuclear cascade. As a result secondary protons, neutrons and pions are generated (Figure 1.4).
Figure 1.4: Spallation reaction due from an impinging particle. [4] In a second stage the excited nucleus resulting from the intranuclear cas-cade undergoes nuclear de-excitation or evaporation" by emitting particles with energy less than 20 MeV, mainly neutrons. For the case of heavy nuclei, high energy ssions can take place as a competitive process to the evapora-tion.
Compared to ssion reactions, spallation mechanism lead to nuclei ca.15 amu lighter than the target. These spallation products are thus closer to the mass number of the original nucleus than in case of ssion processes [4].
Figure 1.5: Mass distribution of debris for 208P b + p. [4]
In terms of released neutron energy, a larger dierence is observed with 25-30 neutrons produced in spallation reactions, with ca.2-3 order of magnitude higher energy than those generated from ssion events.
In a spallation reaction large amount of heat are generated. To remove it a medium with excellent heat transfer properties is needed.
1.4 Lead Bismuth Eutectic as coolant and
spal-lation target
Among the possible candidates coolant materials, from a long time, the choice was between sodium and lead-bismuth eutectic (LBE): both show excellent heat transfer properties as well as low melting points, allowing operation at low temperatures reducing corrosion, and high boiling temperatures, avoiding high pressures.
Table 1.1: Thermo-physical properties of LBE [5]
Properties Units LBE
Atomic number -
-Atomic mass - -
-Melting point ◦C 125
Boiling point ◦C 1670
Mean density solid 20◦C kg/m3 10474
liquid 450◦C 10150
Heat capacity solid 20◦C 0.128
liquid 450◦C kJ/kgK 0.146
Thermal conductivity solid 20◦C W/mK 12.6
liquid 450◦C 14.2
Volume change with melting - % +0.5 Moreover LBE has a low neutron absorption cross section.
Besides all these advantages a major issue arises from the corrosive eects of LBE towards structural materials such as stainless steel. However the use of LBE results in the production of polonium-210 (210P o), due to the neutron
capture reaction of bismuth-209 (209Bi) forming bismuth-210 (210Bi) with
subsequent β-decay. 209 83 Bi + 1 0n −−−−−−−−→ 210 83 Bi 5.3d;β− −−−−−−−−→210 84 P o 138.4d;α −−−−−−−−→206 82 P b (1.4)
One of the most prominent projects of liquid metal-cooled ADS is MYRRHA (Multipurpose Hybrid Research Reactor for High-tech Applications), cur-rently under design at SCK•CEN (Mol, Belgium). It is conceived as a multi-purpose facility consisting of a sub-critical LBE-cooled reactor coupled with a proton accelerator where the LBE is both the coolant and the spallation target.
After 1 year of operation employing a proton beam current of 1 mA, around 2 g of polonium are produced [6]. Polonium 210 is very dangerous for its radiotoxicity and its transport and diusion, in a fast nuclear reactor cooled by LBE, must be accurately analysed.
2
MYRRHA
MYRRHA started as a continuation of the ADONIS (Accelerator Driven Op-timized Nuclear irradiation System) project. The latter consists on a small irradiation facility based on the ADS concept with the solely objective of producing radioisotopes for medical purposes. [7] MYRRHA will be the rst of a kind demonstrator of the transmutation of nuclear waste, thus trans-forming long-lived ssion products into short-lived.
Besides the transmutation of nuclear waste MYRRHA will be used for other applications.
Nuclear medicine gives aids every day to millions of people against the can-cer through the use of radioactive isotopes; 80% of the diagnostic nuclear medical imaging is based on the properties of molybdenum 99mT c. From its
nuclear decay arising from an half life of 6 hours, 99M o is generated and ca.
the 95% of the overall production comes from irradiation of uranium [8] in 5 nuclear sites: Chalk River (US), Petten (the Netherlands), Mol (Belgium), Saclay (France) and Pelindaba (South Africa). The european plants repre-sent ca.40-45% of the overall production of these isotopes [9].
Starting from 2007 some reactor were shut down or went under the ex-tension process: these induced a worldwide need for molybdenum supply; moreover, the end of operation of the BR2 reactor at SCK•CEN is foreseen at the beginning of the 2020's. Therefore, to guarantee the supply of molyb-denum for medical purposes, a replacement for the BR2 is required, hence the multi-purpose nature of MYRRHA.
In addition to the production of medical radioisotopes, MYRRHA is a rst of a kind demonstrator for the transmutation of radioactive waste. Besides these applications, MYRRHA will support other research programmes such as: testing of materials designed for generation IV reactors, fusion technology and space applications, and the production of neutron irradiated silicon for semiconductors currently carried out in the BR2 reactor.
MYRRHA is conceived as a exible fast spectrum neutron reactor. It is composed of 600 MeV proton accelerator which delivers a proton beam to a LBE spallation target, generating the neutrons which sustain the nuclear chain in a sub-critical core cooled by LBE itself. The main consequence of the desired high neutron uxes (and hence the high power density) is the compactness of the core and the small dimensions of the central hole of the spallation target.
2.1 Design
MYRRHA is a pool-type reactor with two vessels: the reactor and the guard vessel. The top space of the reactor vessel, above the LBE surface, the reactor vessel is lled with inert gas (N2). The upper region of the pool is
covered by a steel and concrete plate with the aim, besides support all the instrumentation, of shielding the above regions from the neutrons generated in the core. A scheme of the primary system is shown in Figure (2.1).
Figure 2.1: Detailed MYRRHA reactor cross-section of the primary system. (A) Reactor vessel; (B) reactor cover; (C) diaphragm; (D) primary heat
exchanger; (E) primary pump; (F) in-vessel fuel handling machine; (G) core; (H) above core structure and (I) core restraint system.
The so-called diaphragm separates the hot and cold zones. All primary systems (heat exchangers, fuel handling machine tools and in-vessel inspec-tion instruments) are included in the reactor vessel.
Primary cooling system
The primary cooling system consists of two primary pumps and four heat ex-changers. In order to cool the LBE, it is pumped from the hot (300 − 400◦C)
to the cold (200◦C) zone through heat exchangers. Natural circulation,
de-sirable in case of emergency, is ensured due to the height dierence between the hot and cold regions.
Figure 2.2: (a) Pump, (b) Heat exchanger and (c) their coupling. [10] [11] The primary heat exchangers use pressurized water as secondary coolant. Therefore, in case of rupture of one of the tubes of the heat exchanger, steam would leak leading to its contact with LBE. To avoid this scenario double walls are foreseen in the current design. Besides the primary cooling sys-tem two independent secondary cooling syssys-tems, each comprising two heat exchangers and a steam separator, are foreseen. The secondary circuit, with-out any pumps, is a passive system whose operation is ensured by natural circulation due to the hydrostatic head dierence between the hot (heat ex-changer) and the cold leg (steam separator).
The sub-critical core is formed by 151 hexagonal channels, 69 of which are loaded with fuel assemblies, with MOX fuel characterised by a Pu enrich-ment up to 30%. The level of sub-criticality is 0,955 leaving a safety margin in case of positive increases of reactivity.
The fuel bundle consists of 90 fuel rods and the last dummy element in the center of the bundle for instrumentation and monitoring [12].
Figure 2.3: MYRRHA Fuel Assembly. [13]
The cladding material for the rst MYRRHA cores will be 15-15 Ti ASS (austenitic stainless steel). Advanced ferritic-martensitic steel (FMS) T91 would be used once quali cation would be completed owing to its lowered embrittlement under irradiation above 350◦C. Some alternative solutions are
AISI 316, Ti and AISI 316 L.
Figure 2.4: core con guration. [10]
The central position of the core is reserved for the spallation target mod-ule.
2.2 Proton accelerator
The accelerator is among the most critical parts, from an engineering point of view. It consists of a linear accelerator (LINAC) delivering a continuous proton beam of up to 4 mA at an energy of 600 MeV. The LINAC consists
exclusively of a series of superconducting cavities (made of niobium). The lowest energy part of the LINAC, in which beam losses are unavoidable, is de nitely normal conducting (made of copper). The superconducting LINAC for MYRRHA has 2 sections, the front end" and the independently phased superconducting section". The transition between the sections is placed at 17 MeV.
Figure 2.5: Linac layout. [14]
The front end" starts with an Electron Cyclotron Resonance (ECR)ion source, whose technology has been widely investigated and veri ed at higher beam currents than the one of MYRRHA, thus providing a relatively large margin for further reliability improvements.
The "independently phased superconducting" part is the section which increases the energy of the particles from 17 MeV up to the nal value of 600 MeV. This increase is achieved with the adoption of spoke cavities till 90 Mev while the use of elliptical ones is foreseen at higher values.
The use of superconducting cavities is based on three main technological considerations. The achievable accelerating elds are limited by the ohmic losses. Hence, due to their lowered values in the superconductors, higher elds can be obtained, compared to the normal conducting materials.
The need of converting the energy of the incident beam into secondary particles, instead of heating the target, led to the choice of protons as particles of the beam.
First of all we need to explain what the stopping power is. The proton stopping power can be calculate as follows, with in uence of several factors such as the charge, energy and mass of the interested ion used as projectile. [4]
Si = Zi2Sproton
Ei
Ai (2.1)
where:
Ei ion kinetic energy;
Ai ion mass;
Sproton proton stopping power;
Zi ion charge.
When collisions take place, the neutron production is evaluated by the so-called neutron yield Yi.
This neutron yield is dierent depending on the incident particle in the target (Figure 2.6).
As it can be deduced from Figure (2.6), protons are among the best choices together with deuterons. Empirically the energy dependence of the neutron production can be expressed by the following formula [4]:
Nneutron
proton = M0+ M1E
x
proton (2.2)
where:
x, M0, M1 are tabulated data and the ratio Nprotonneutron expresses the number of
neutrons per incident proton.
The maximum neutron production per incident proton is located at a proton energy of ca. 0, 8 − 1GeV . Beyond this threshold it decreases slowly, while for lower proton energies the decrease is faster due to the preponder-ance of pion production over that of neutrons.
2.3 Spallation loop
The spallation loop can be regarded as the interface between the accelerator and the sub-critical core. In this zone the spallation reaction takes place giving rise to the required neutron ux (2 · 107n/cm2s) with a heat power
deposition of about 2 MW in the target.
Two designs are proposed for the spallation loop: window" or windowless" spallation targets.
2.3.1 Window spallation target
It is the simplest con guration of the spallation loop with a physical bound-ary, the beam tube, separating the target material from the vacuum of the accelerator's line. The window is placed at half the height of the core and cooled by convection exchange with an upward ow of LBE, pumped by means of the primary pumps.
Figure 2.7: Window-type spallation target. [11]
The main issue within this con guration lies on the heavy energy or heat loads to which the nal part of the beam tube, the beam window, is subjected. This fact obliges to regular and not lasting replacements.
2.3.2 Windowless spallation target
Owing this issues an alternative design, the windowless spallation target, was proposed to avoid structural problems on the path of the high intensity
proton beam. The term windowless" refers to the absence of a physical barrier between the target material and the accelerator's line.
A funnel-shaped nozzle allows to have a free surface at the end of the beam tube, at the interface between the vacuum of the accelerator's line and the target material (Figure 2.8). It is foreseen a detachment of the LBE ow from the wall oering an advantage in case of volume changes when the proton beam is switched on or o. [15] [16]
Figure 2.8: LBE ows by gravity in the annulus around the beam tube; the proton beam hits on the LBE jet once it overpasses the end of the tube. [16] All the components of this spallation loop, i.e. pump, heat exchanger and nozzle, form a closed loop so that all the products generated as a product of the spallation reaction are kept isolated from the primary system.
Figure 2.9: Windowless-type spallation target. [16]
A service vessel is used to contain all the main parts of the spallation loop. This vessel is placed near the border of the reactor vessel, housing the pump, the system for the control of the oxygen, the instrumentation and the vacuum system.
3
Polonium issue in MYRRHA
As already mentioned the use of LBE as coolant in MYRRHA is supported by several thermodynamics properties (i.e. low melting and high boiling points). However it presents certain chemistry challenges. Among them the formation of hazardous radionuclides such as Hg and Po, by various spallation and capture reactions, is of primary importance. Speci cally the majority of polonium is in the form of Po-210 due to the neutron capture reaction from bismuth (Equation 3.1). Some additional small quantities are generated as a result of the interaction of the proton beam with the LBE spallation target (Equation 3.2). 209 83 Bi + 1 0n −−−−−−−−→ 210 83 Bi 5.3d;β− −−−−−−−−→210 84 P o 138.4d;α −−−−−−−−→206 82 P b (3.1) 209 83 Bi + 1 1p −−−−−−−−→ 210 84 P o + xn (3.2)
Po-210 has an half life of ca.138 days, with a large decay heat (140W/g) [17]. Moreover polonium toxicity (25000 times more than hydrogen cyanide [18]) imposes severe limits on the quantities allowed to be managed and delivered. Therefore general explanation of the thermodynamics of evaporation of dis-solved components is presented in the following Section. Polonium belongs to the Chalcogen group together with Selenium, Tellurium, Sulfur and Oxy-gen. As a result of these reactions, after 2 years of operation, ca.2 kg of Po will be dissolved in LBE (corresponding to a mole fraction of 10−7). Under
certain conditions polonium may be released from its solution. Due to its elevated radiotoxicity and its tendency to evaporate, a qualitative and quan-titative evaluation of Po evaporation from LBE is required for the design, safety analysis and licensing of LBE-based nuclear installations.
3.1 Evaporation of dissolved components
Evaporation is normally characterized by the vapour pressure of a component in solution. This vapour pressure is de ned as the pressure exerted by a vapour when is in equilibrium with the liquid or solid form, or both, of the same substance [19]. In the case of a solution of solute (A) and a solvent (B), the vapour pressure of each components Moreover the partial pressure is de ned as the pressure each gas of a mixture would exert if it alone would occupy the whole volume under the same conditions. [20]. Ideally, assuming that the bonding energies between A and B molecules are equal, the solution follows the Raoult's law and the vapour pressure of the solute A can be calculated by:
PA = PA0XA (3.3)
where:
PA0 is the vapour pressure of the pure element; XA is the molar fraction of the solute A.
This approximation, even though it oers a good level of accuracy in highly dilute solutions and for components with similar characteristics, e.g. polonium in LBE, is however not representative of real situations.
In non ideal solutions the bonding energies AA and AB are dierent. There-fore, to account for these deviations, the vapour pressure is calculated from the thermodynamic activity aA. The thermodynamic is then related with
the mole fraction XA by:
aA= XAγa (3.4)
where:
γa is the thermodynamic activity coe cient, which expresses the deviation
from ideal behaviour in the interactions.
Thus the vapour pressure of A in solution is calculated by:
PA= PA0aA (3.5)
In the speci c case of very dilute solutions with low XA, γAtends to be
inde-pendent from the mole fraction. Under these conditions the vapour pressure of a solute is normally described by the Henry's Law:
PA= KH,AXA (3.6)
where:
KH,A is the Henry constant [Pa].
This Henry constant can be regarded as the vapour pressure of a pure solute solution at standard conditions (Figure 3.1).
Figure 3.1: Applicability of Henry's and Raoult's Law [21].
Generally the relation between the vapour pressure or Henry constant, de-pending on the dilution levels, with temperature is described by an Antoine-type correlation:
KH,i= 10
A
T+B (3.7)
where A and B are tabulated.
In the speci c case of Po evaporating from LBE in MYRRHA, the Henry's law is expected to be applicable owing to the low mole fractions expected in the reactor (ca. 10−7). The Henry constant would then describe the
equilibrium constant of the evaporation process:
P o(LBE) *) P o(g) (3.8) The relation between the Gibbs free energy of a reaction at any moment and the standard state is given by:
∆G = ∆G0+ QRT (3.9)
where:
Qis the reaction quotient.
The equilibrium condition means that there is no driving force in the reaction, i.e. ∆G = 0, and Q is called the equilibrium constant. In this particular case, being the Henry constant the equilibrium constant:
KP o(LBE)= exp(−
∆G0
where the Gibbs free energy at standard conditions ∆G0 is de ned as:
∆G0 = ∆H0− T ∆S0 (3.11)
where:
∆S0 is the entropy variation [Jmol−1K−1];
∆H0 is the enthalpy variation [Jmol−1].
From Equation (3.10) it is possible to express the Henry constant as follows [22]: KP o(LBE)= 10 ∆S0 2.303R− ∆H0 2.303RT (3.12)
From a comparison of Equation 3.7 and Equation 3.12 the expressions for the coe cient A and B are evaluated:
∆S0 = 2.303R(A − 5) ∆H0 = −2.303RB
eight of the substance.
In the following sections we will focus on the available literature data on metallic Po evaporation and that of Po from LBE.
Vapour pressure of polonium
The tendency of Po to evaporate is described by its elevated vapour pressure. The rst investigations on the evaporation of metallic Po were performed by L.S.Brooks [23]. Evaporation experiments between 438◦
Cand 735◦C allowed the derivation of a vapour pressure-temperature correlation:
PP o0 = 10−5377.8±6.7T +9.3594±0.0068 [438 ÷ 745◦C] (3.13)
In more recent study on the evaporation of metallic polonium, although al-ready performed some decades ago, Abakumov et al. [24] determined the vapour pressure of Po, assuming that it evaporates as monoatomic species, between 368◦C and 604◦C:
PP o0 = 10−5440±60T +9,46±0,05 [368 ÷ 604◦C] (3.14)
Added to these experimentally determined vapour pressure-temperature cor-relations, theoretical estimations were carried out by B. Eichler [25]. The estimation of the vapour pressure of monoatomic and diatomic Po relied on the extrapolation of thermodynamic properties such as enthalpy or entropy of its lighter homologous, i.e. S, Se and Te. When comparing experimen-tal and theoretical estimations a signi cant deviation among them could be observed.
Figure 3.2: Comparison between experiments and theoretical values. Deviations between experimental results (squares and triangles) and extrapolated (circles) are larger at low temperatures. (liquid-solid
transition is shown by an arrow) [17]
In Figure (3.2) it is shown that the dierence between the experimental values and the theoretical estimations by Eichler et al. is more signi cant at lower temperatures. Thermal eects derived from the Po decay were suggested as a plausible contributor to the observed dierences [25].
In addition another factor might be the considered gaseous compound. Abakumov already suggested the presence of a small percentage of polonium molecules in their diatomic form. The theoretical estimations on the other hand revealed the dominance of the diatomic Po in the gas phase in contact with pure samples. The monoatomic species would be preponderant in highly dilute solutions of Po. Therefore, owing to the good agreement between experimental and theoretical data when P o2(g)is considered the evaporating
species (Figure 3.2), it can be a rmed that the theoretical estimations are reasonable.
3.2 Transpiration method
There are several methods to measure the vapour pressure of dierent sub-stances. These methods may be classi ed as: static method, boiling point method, eusion method and transpiration method. The latter, also known as transportation method, was the chosen method to characterize Po evapo-ration from LBE in the last few decades.
3.2.1 Fundamentals
In the transpiration method a carrier gas is sent over a sample at a cer-tain ow rate that ensures the establishment of equilibrium conditions. This owing gas will transport away the vapours emanating from the sample when heated to a certain temperature. These vapours condense later downstream the sample in colder regions of the setup. The quanti cation of these vapours allows the determination of the vapour pressure of the studied substance [26]. A scheme of a typical transpiration apparatus is shown in Figure (3.3). The gas enters the setup (A in gure) and it ows over the sample placed in an isothermal region of the apparatus (zone C). The released vapours then condense in colder regions of the setup (zone E). The constrictions B and D are used to minimize diusion losses during the experiments to ensure the realization of saturation conditions in the sample region.
Figure 3.3: Scheme of transpiration apparatus. See text for further explanations [26].
The most commonly used materials in transpiration apparatus are glass or quartz. Argon, helium, nitrogen and hydrogen as carrier gases. Once the carrier gas is own through the outlet for the desired time transient, some measurements take place depending on the nature of the investigated ele-ment: they can be chemical or radiometric if there is the need to analyse radioactive isotopes.
The most crucial part of using the transpiration method to measure vapour pressures is the achievement of saturation conditions within the tested con-ditions. To this aim vapour pressure data should be collected in a region of ow rate where the carrier gas is saturated with the sample vapours and where the transport of these vapours occurs mainly by convection. These conditions are normally ful lled at intermediate ow rates when the vapour pressure is independent from the carrier gas ow rate (see Figure 3.4). How-ever when the gas ow rates are lower or higher these conditions are not reached.
Figure 3.4: Variation with ow rate of the vapour pressure in the carrier gas at the outlet (reproduced from [26]
When looking at release amount, an alternative way of visualizing it can be found in Figure (3.5).
Figure 3.5: Evaporation rate in the carrier gas variable with the ow rate. See the text for further explanations. [27]
At low ow rates, the contribution of the diusion term is not negligible when compared to convective transport and leads to an overestimation of the vapour pressure values. On the other hand, at higher values of gas ow rate, the emanating vapours are transported away from the sample position before equilibrium could be established, resulting in an underestimation of the vapour pressure.
Besides the achievement of the equilibrium conditions, the understanding of the evaporation of polonium dissolved in LBE requires a deeper insight in physical processes such as diusion of polonium towards the surface from the bulk of the LBE, evaporation to the gas phase above the liquid-gas interface and diusion of Po in the carrier gas followed by its transport by convection.
3.2.2 Po diusion in LBE
The diusion process of solutes in liquids can be one of the limiting steps subsequent for evaporation processes due to its slowness. There are plenty of examples in natural and industrial activities as in case of digestion or during corrosion processes where diusion controls their rate. Owing to its slow rate, the evaluation of the diusion of the solute in the solvent is crucial. In the speci c case of polonium in LBE the diusion through the LBE sample can be described by the second Fick's law:
∂CP o(LBE) ∂t = DP o(lBE) ∂2C P o(LBE) ∂y2 (3.15) where: ∂CP o(LBE)
∂t is the variation in time of the concentration of solute, speci cally
polonium, in the solvent, LBE;
DP o(lBE) is the diusion coe cient [m2/s]; ∂2C
P o(LBE)
∂y2 is the curvature of the concentration's pro le.
Generally the boundary and initial conditions to be applied to such a problem are: CP o(LBE)= CP obulk|t=0 y ∈ (−∞; 0] CP o(LBE)= CP obulk|t>0 y → ∞
CP o(LBE)= CP osurf ace(t)|t>0 y = 0(surf ace)
(3.16) This set of conditions is valid for a liquid domain that can be assumed semi-in nite, but in transpiration experiments with relatively small samples, this hypothesis is no longer valid and the following conditions have to be adopted:
(
CP o(LBE)= CP obulk|t=0 y ∈ [ybottom; 0] ∂CP o
∂y |t>0 = 0 y = ybottom
(3.17) If Po diusion in the melt is slow with a limited convective motion, Po depletion near the surface might occur. The condition would result in the hindrance of the polonium evaporation from LBE [28]. However if diusive motion is faster than the other involved phenomena, an enhancement in the exchange rate at the liquid-gas interface occurs. To describe the diusive behaviour of polonium, once it is dissolved in LBE, an evaluation of its diusion coe cient is required. To this aim, three dierent approaches of evaluating are used. The Stokes-Einstein, the Roy-Chhabra and the Wike-Chang formula.
Stokes-Einstein
Even though its accuracy is not the best (i.e. 20% [29]), the most common way to evaluate a diusion coe cient in a liquid medium is the adoption of the Stokes-Einstein equation.
The derivation of this model assumes that the solute molecule is a sphere in motion inside a still solvent, thus not considering any relative motion between the two bodies. The forces acting on the sphere's surface are supposed to be directly proportional to the velocity vsphere following the Stoke's law:
f orce = 6πR0µvsphere
with:
µthe viscosity R0 the solute radius
The acting force on the sphere of solute may also be represented as:
f orce = −∇µsolute (3.18)
The gradient of the chemical potential was de ned as: ∇µsolute =
kBT
csolute
∇csolute (3.19)
where, therefore, rearranging the previous equations: − kBT csolute ∇csolute = 6πR0µvsphere (3.20) csolute· vsphere = − kBT 6πR0µ ∇csolute (3.21)
Finally considering that the current from the Fick's law is de ned as: J = csolute· v = D∇csolute (3.22)
the diusion coe cient can be determined combining Eqs 3.21 and 3.22: Dsolute(solvent) =
kBT
6πµsoluteR0
(3.23) where:
Roy-Chhabra
Roy and Chhabra on the other side determined diusion coe cients of solutes in liquid metals assuming that the diusion of the solute only depends on the dimensions of the involved species, ignoring the in uence of the interactions between the dierent molecules. This assumption yielded an accurate estima-tion of the diusion coe cients of dierent solutes in liquid metals. Therefore the diusion coe cient of polonium in LBE may also be calculated by [30]:
DP o,LBE = 0, 2BRd3 LBE dP o T VLBE VLBE− V0 V0 (3.24) where:
dP o, dLBE are the atomic diameter of polonium (0.197nm)and LBE (0.205nm);
B is a typical parameter of the mixture; VLBE is the molal volume of the liquid metal.
V0 is the so-called close-packed molal volume given from [31]:
1 η = B[
VLBE
V0− 1
] (3.25)
where η is the viscosity of the liquid metal. Wike-Chang
Added to the previous expressions to evaluate the diusion of Po in LBE, an additional method is based on the Wike-Chang equation . This approach is valid when solute and solvent have similar dimension [29]:
DP o(LBE)= 7.4 · 10−8T√φMLBE µV0.6 P o (3.26) where:
φ is an empirical parameter (2,6 for water, 1 for organic solvents and 1,5 for alcohols);
MLBE is the molecular weight of LBE [Da]. Dalton is the atomic mass unit
representing the mass on molecular or atomic scale: 1Da = 1.660538·10−27kg;
VP o is the molecular volume of polonium.
Owing to the absence of data on liquid polonium viscosity and the em-pirical parameter φ, this formula might not be applicable to the case of Po diusion in LBE.
3.2.3 Evaporation
Assuming the realization of saturation conditions and the transport of vapours mainly by convection in the investigated conditions, a mass balance in the region above the samples allows the determination of the amount of Po evap-orating from the LBE sample:
dnP o
dt = − ˙V csat (3.27) where csat is is the concentration of vapours above the LBE sample when
saturation conditions are reached.
If vapours are assumed to behave as an ideal gas: nP o(t) = nP o(0) −
˙ V psat
RT t (3.28)
where:
R is the gas constant; T is the temperature in [K].
In the speci c case of Po evaporating from LBE, with a Po mole fraction of ca.10−7, the saturation pressure can be derived using Henry's law:
psat = KP o(LBE)
nP o(t)
ntot(t)
(3.29) where ntot = nP o+ nLBE.
Due to the small amounts of Po dissolved in the LBE, ntot can be
approx-imated to the weight of LBE sample:
ntot(t) = nLBE + nP o(t) ≈ nLBE (3.30)
Therefore rearranging Equation 3.27 and integrating over the experiment time we obtain: nP o(t) = − Z V K˙ P o(LBE) RT nP o(t) nLBE dt (3.31) so: ln[nP o(t)] = − ˙ V KP o(LBE) RT nLBE t + C (3.32)
Considering the initial condition n(0), the value of the integration constant can be calculated. Hence Equation (3.32) is then transformed into:
nP o(t) = n(0) exp − ˙ V KP o(LBE) RT nLBE t ! (3.33)
The Henry constant describing the evaporation process of Equation (3.8) can then be obtained by:
KP o(LBE)= − RT nLBE ˙ V t ln nP o(t) n(0) (3.34) In the speci c case of experiments where the amount of evaporated solvent (LBE in this case) in the solution is small:
exp " −V K˙ P o(LBE) RT nLBE t # → 0 (3.35)
So Equation (3.33) can be rearranged as [28]: n(t)
n(0) = − ˙
V KP o(LBE)
RT nLBE (3.36)
3.2.4 Diusion of Po in carrier gas
Once Po evaporates from the sample it mixes with the carrier gas owing above the sample. The extent of this mixing is de ned by the diusion of Po in Argon. This diusion coe cient was estimated using Chapman-Enskog equation [29]. This theory provides coe cients with an accuracy of ±8%, and it assumes that each interaction between gaseous compounds involves only two molecules.
The derivation of the Chapman-Enskog equation assumes that the gas ow is composed by a multitude of rigid spheres; its diusive ow could be evaluated by: n = −1 3νl dc dz + c1ν 0 (3.37)
where the term −1
3νlrepresents the diusive motion, resulting from the
prod-uct of the average molecular velocity ν de ned as: ν =p2kBT /m
with m the molecular mass and l the mean free path, while the term c1ν0
represents the convective transport.
When comparing Equation (3.37) with Fick's law: D = νl/3
Where the mean free path l is: l = 4kBT /pπσ2
with the numerator being the concentration of molecules per unit of volume [29], and σ is the molecular diameter. From the latter three expressions the diusion coe cient can be written as:
D = r 2kBT m 4kBT 3pπσ2 = 4√2 3π ! (kBT )3/2 √ mpσ2 (3.38)
The improvement of the Chapman-Enskog theory over the previous one is the direct inclusion also of the collisions' details through the Ω parameter:
DP o,Ar = 1.86 · 10−3T3/2p(1/M P o+ 1/MAr) pσP oAr2Ω (3.39) where:
MP o, MAr are the molar weights of polonium and argon;
pis the total pressure [atm];
σP oAr is the collision diameter [Å] estimated as average of the diameters of
the two elements;
Ω is a dimensionless constant which depends on the energy of interactions 12 with: 12= √ 12 where 12 = √
12 are tabulated data.
If the Chapman-Enskog equation (Equation 3.39) is compared with the one of the general theory (Equation 3.38), the same dependency on temper-ature, pressure and material properties are present in both the approaches, but the Chapman-Enskog's presents the advantage of including the contri-bution of the interaction between molecules via the Ω parameter that takes into account the energy of interaction between particles.
3.3 Po evaporation from LBE
In the last decade most investigations of Po evaporation from LBE have been carried out resulting to the transpiration method. Firstly Neuhausen et al. studied the short (ca. 1 h) and long (10 to 28 days) term evaporation behaviour of polonium from LBE. The results of the short-term experiments showed no measurable Po release below 700◦C, while the temperature needed
to evaporate 50% of the element was 927◦C. On the other hand in the
long-term evaporation experiments at 594◦C an evaporation rate of the order of
1%per day was measured. No measurable release was detected below 594◦C. Using the combined results a Henry constant-temperature correlation was derived, valid in the temperature range between 700 and 1057◦C.
logKP o(LBE) = −
8592
T + 10.2 [700 ÷ 1057
◦
C] (3.40) Until this point no dedicated experiments were carried out to verify whether these data were collected under equilibrium conditions. To check whether saturation conditions were realized, Ohno et al. [32] conducted several ow rate-dependent evaporation experiments in the range between 30 and 120 cm3/min at temperatures between 550 and 750 ◦C using Po-doped LBE
samples with a Po mole fraction of 2 · 10−10. Tests performed for 210P o at
450◦C in the ow rate range between 60 and 120 cm3/minrevealed a similar behaviour. Therefore this evaporation setup was further used to investigate the simultaneous LBE and Po evaporation between 450 and 750 ◦C under
Ar carrier gas. A temperature correlation for the Henry constant was then derived for Po evaporating from LBE:
logKP o(LBE)= − 8348 T + 10.5357 [450 ÷ 750 ◦ C] (3.41)
Experiments at SCK•CEN
In the last four years, within the MYRRHA R&D project, the evaporation behaviour of Po from LBE was studied at SCK•CEN.
In a rst experimental campaign Po-doped samples were heated between 600◦C and 1000◦C and the amount of the released Po was measured.
Po-doped LBE samples were in the of the BR-1 reactor by neutron irradia-tion, resulting in samples with a Po mole fraction of 10−10.
Once the samples were ready, they were inserted in the quartz tube above a molybdenum foil; afterwards the setup was heated up with a tubular fur-nace.
The experiments were carried on transpiration setup like the one shown in Figure (3.6). In this experiment, a carrier gas (Argon) is sent inside a quartz
tube containing a Po-doped LBE sample at a certain ow rate that ensures the establishment of equilibrium conditions.
The LBE sample is placed in a region of the tube where an oven melts it and ensures the isothermal conditions.
Figure 3.6: Apparatus for transpiration method at SCK•CEN. [22] The dimensions of sample and setup, and experimental conditions are listed in Table 3.1 [22] [33].
Table 3.1: Geometrical parameters of the transpiration experiment. [33] Sample diameter (before melting) [mm] 5
Sample length [mm] 13
Outer tube diameter [mm] 10 Outer tube length [mm] 450 Estimated evaporation surface [mm2] 60
Initial Po molar fraction [−] 10−10− 10−12
Carrier gas - Argon/5%H2
Pressure [bar] 1
Flow rate [ml/min] 20-300
Experiment time [s] 3600
To avoid polonium escaping from the setup, a set of lters was installed downstream the quartz tube.
The amount of Po released from the sample was quanti ed by collecting the vapours condensed on the quartz tube walls during evaporation experiments. To this aim, the tube was rinsed twice with a 7M HNO3(aq) solution. The
rinsing fractions were subsequently mixed with scintillation cocktail and mea-sured by liquid scintillation counting. In a rst experimental campaign, with tests performed between 600◦Cand 1000◦C, the methods were validated with
simultaneous LBE and Po evaporation experiments [22].
Subsequently the achievement of saturation conditions and dominant con-vective transport of vapours were demonstrated by ow rate-dependent ex-periments. The results revealed the realization of equilibrium conditions in the ow velocity region between 5cm/s and 10cm/s (Figure 3.7), when the Henry constant was independent from the ow rate. Moreover, at lower and higher ow rates the behaviour predicted in Ref. [27] was satisfactorily reproduced.
Figure 3.7: Experimental values of Henry constant at 600◦C and 1000◦C
. [33]
For ow velocities lower than 5cm/s the Henry constant values were over-estimated due to the not negligible contribution of diusion of polonium in the gas phase. The driving force for diusive transport at low ow rates is the concentration gradient between the gas above sample and the further regions of the tube.
At higher velocities there is a decrease in the Henry constant values due to unsuccessful establishment of equilibrium condition.
Besides the determination of equilibrium properties (Henry constant), these ow rate-dependent evaporation experiments allowed the derivation of other properties.
Among them, a plot of the release rate trend as function of the ow velocity (Figure 3.7) revealed a constant region at velocities higher than 10cm/s and
14cm/sfor experiments performed at 600◦C and 1000◦C respectively. Under these conditions the release of Po from the LBE surface reaches the maximum value [33].
Figure 3.8: Experimental values of release rates at 600◦C and 1000◦C . [33]
This evaporation rate was then estimated through the following formula [33]: 1 nP o(LBE) dnP o(LBE) dt = αP o(LBE)A KP o(LBE) √ 2T0RMP oπ (3.42) where:
αP o(LBE) is the vaporization coe cient of polonium, approximately 10−3;
MP o= 210g/mol is the molar mass of polonium;
A is the exchange area estimated to be 60mm2.
Even though already observed in experiments, the validation of the trends shown in Figs. (3.7) and (3.8) through computational modelling, would con- rm the realization of saturation conditions in the ow velocity region be-tween 5 and 10 cm/s.
Therefore, the main objective of this work consists on studying this ow rate dependence of Po evaporation from LBE, and helpfully reproduce the trends observed in Figs. (3.7) and (3.8) in all the studied ow rate regions.
4
Mathematical models
The program used to simulate the experimental test of transpiration is ANSYS-CFX, a computational code able to model single and multiphase ows. It is helpful to brie y introduce the equations and the implemented models for the resolution of its calculations. The governing equations are the unsteady state balance equations for the mass (continuity equation), momentum and energy, without considering the eects of combustion or radiation.
The set of equations is averaged in time, introducing terms that take into ac-count oscillations due to the turbulence nature of the ow; then it is required the addiction of relations, by means of turbulence models, to close the math-ematical problem. The thermodynamic properties of the involved chemical elements are inserted in the proper section of the setup as well as the adopted expressions for all the parameters responsible for the mass exchange [34].
4.1 Governing equations
The set of coupled partial dierential equations, describes how the velocity, pressure, temperature and density are related inside a uid in motion. They are based on the Euler equations.
The rst is the time dependent conservation of mass: ∂ρ
∂t + ∇ · ρ ~w = 0 (4.1) where:
∂ρ
∂t represents the time variation of the mass;
∇ · ρ ~w is the advective term responsible of the mass transport in the cell. The second is the time dependent conservation of momentum (Navier-Stokes equation):
∂ρ ~w
where:
∂ρ ~w
∂t represents the time variation of momentum;
∇(ρ ~w · ~w)is the advective term of the momentum transport in the cell, also called Reynolds stress tensor;
τ is the stress tensor and it represents the inertial term; ∇prepresents the eects of the pressure forces;
SM represents a source term of momentum.
τ expresses the relation between the stress and strain tensor, which is sup-posed to be linear and its complete form is [35]:
τi,j = 2µei,j+ [µ
0
− 2
3µ](∇ · ~w)δi,j Generally the bulk viscosity µ0
is neglected, leading to a simpler formulation: τ = µ
∇ ~w + (∇ ~w)T − 2 3δ∇ ~w
Finally, the third equation is the time dependent balance of energy: ∂ρhtot ∂t − ∂p ∂t + ∇ · (ρ ~whtot) = ∇ · (λ∆T ) + ∇ · ( ~w · τ ) + ~w · SM + SE (4.3) where: htot = h(T, p) + 1 2w
2 is the total enthalpy, depending on the static one; ∂ρhtot
∂t is the time variation of thermal energy; ∂p
∂t is the term responsible for the work of the pressure forces;
∇ · (ρ ~whtot)is the advective term of the thermal energy transport in the cell;
∇ · (λ∆T )is the contribution of the conductive thermal ux, neglecting the thermal radiation ux;
∇ · ( ~w · τ ) is the mechanical power dissipated by friction forces; ~
w · SM is the work due to external sources;
4.2 Turbulence models
Turbulence ows are characterised by uctuations of the main variables, like pressure, temperature and velocity, around average values, also able to vary in time. In case time scales of turbulent phenomena is much smaller than the ones typical of the transient, the exact description of their temporal evolution is of limited interest; then a statistical treatment over these dimensions is generally adopted.
The time (Reynolds) average of the variable c leads to a mean and uctuating component: c(t) = ¯c + ∆c(t) = 1 ∆t Z ∆t/2 −∆t/2 c(τ )dτ + ∆c(t) (4.4) where ∆c(t) has null mean value: R∆t/2
−∆t/2∆c(τ )dτ = 0. From averaging the
momentum equation over the time, we get: ∂ρ ¯w~
∂t + ∇(ρ ¯w · ¯~ w) = ∇ · ¯~ τ − ∇¯p + SM (4.5) ρ ~w ~w is the averaged Reynolds stress tensor, formed by 9 unknowns that reduce to 6 thanks to its symmetry.
Its resolution is the focal point of every RANS (Reynolds Averaged Navier Stokes equations) analysis and several approaches can be followed.
On one hand there is the adoption of the Boussinesq hypothesis, on the other hand the direct solution of the 6 component of the tensor.
4.2.1 Boussinesq hypothesis
This approach makes use of the Boussinesq assumption: the Reynolds stresses are considered to be proportional to the mean velocity gradient, in analogy with a Newtonian uid in laminar ow; hence the de nition of the turbulent (or eddy) viscosity.
ρ ~w ~w = µT ∂ ¯wi ∂xj + ∂ ¯wj ∂xi (4.6) There are several models to solve the tensor, based on increasing number of equations that have to be added to the original set of balance equations. They dierently calculate the value of the eddy viscosity, to evaluate the Reynolds stress tensor and hence the contribution of the turbulence.
Zero-equations models
The so called zero-equations" models use an algebraic relation involving the Prandtl mixing length to calculate µT:
µT = ρl2T ∂wi ∂xj (4.7) where:
lT is the Prandtl's mixing length.
ANSYS implements the zero-equations model using a similar relation [34]: µT = ρlTf WT (4.8) where: lT = 1 7V 1/3 D
WT is the maximum value of velocity in the uid domain.
Besides the zero-equation models, there are one or two-equation turbu-lence models; in this way the eddy viscosity is evaluated through the relation between two variables, the turbulent kinetic energy k and the turbulence kinetic energy dissipation ε = −dk/dt:
µT = Cρ
k2
ε (4.9)
The values of k and ε come from their respective transport equations. One-equation models
This model takes the name from the adoption of one transport equation for the kinetic turbulent energy, de ning the turbulence kinetic energy dissipa-tion as [36]:
ε = Ck
3/2
l
Even though one-equation models are incomplete due to the need of adapting the length scale l case by case, ANSYS is equipped with a simple version developed by Menter [34].
Two-equations models
These models use equations for transport of two of the following variables: k, ε, ω and τ, where omega is the dissipation rate and tau is the turbulence dissipation time.
ω = ε k
k − ε model This is the most widely used two equation turbulence model coming from the work of Daly and Harlow (1970) and from Launder and Spalding (1972) [37]. Two transport equations for k and ε are solved.
∂ρk ∂t + ∇ · (ρk ~w) = ∇ · µ + µT σk ∇k + Pk− ρε ∂ρε ∂t + ∇ · (ρε ~w) = ∇ · µ + µT σε ∇ε + ε k(C1Pk− C2ρε) (4.10) Then the eddy viscosity is evaluated through the following relation:
µT = Cρ
k2
ε (4.11)
The resolution of the turbulence phenomena near the wall is di cult due to the high shaped gradient of the turbulence intensity. Owing to this fact, the choice is whether to perform a low-Reynolds simulation with a very re ned mesh, or to not simulate the boundary layer with the adoption of a wall function that puts a prede ned velocity pro le near the wall.
In both the cases the qualitative analysis of the mesh is based on the non-dimensional distance from the wall:
y+= y Iv
ν
If there is the need to solve a low Reynolds model, the rst layers of the mesh have to be very re ned, in particular the rst one to get an y+ << 1. The
applicability of the k − ε model requires an y+< 0, 2, though ANSYS puts a
wall function below the limit of y+ = 11, 06 to avoid the user commit errors
in getting mesh of that grade of re nement.
k − ω Wilcox model Contrary to the previous model, the Wilcox model does not have a formal derivation of the transport equation for ω, the dissi-pation rate as de ned by Kolmogorov in 1942:
ω = ck
1/2
l
where c is a constant. The set of two equation is:
∂ρk ∂t + ∇ · (ρk ~w) = ∇ · µ + µT σk ∇k + Pk− β0ρω ∂ρω ∂t + ∇ · (ρω ~w) = ∇ · µ + µT σω ∇ω +αω k Pk− βρω 2 (4.12)
where:
Pk is the production rate of turbulence.
Once k and ω are known, the eddy viscosity is evaluated this way: µT =
ρk
ω (4.13)
The advantage of a k − ω model, compared to a k − ε, is the near wall treat-ment: while in a low Reynolds simulation with k −ε an y+ < 0, 2is required,
here y+< 2 is enough [34].
Moreover ANSYS-CFX does not have implemented a Low-Reynolds treat-ment in the k − ε model.
ω-based SST model The Shear Stress Transport model is again a blend of the k − ω Wilcox model and the standard k − ε. The drawback is the overestimation of the eddy viscosity, hence the idea of impose a limitation on it:
µT = ρ
ak max(aω; SF2)
where:
F2 is a blending function used to limit the value of µT in the vicinity of a
wall; S = 1 2( ∂wi ∂xj + ∂wj
∂xi)is the strain rate.
4.2.2 Stress Transport models
The other strategy to solve the Reynolds stress tensor is without the assump-tion of a turbulent viscosity, but solving six transport equaassump-tions, one for each of the tensor's component.
This method should be more suitable for ows where some anisotropies are expected [34], with the drawback of di culties in its convergence during the calculation due to the higher complexity of the calculations.
Inside the terms of the stress transport equation, the turbulence dissipation term ε appears, hence it is required one more equation for this quantity to close the problem.
ω-based BSL model This model follows the general scheme of the two-equations BSL model: it adds an equation for ω and the coe cients of the calculation are evaluated by switching from an ε form (expressed in the way of kω) in the bulk, to the ω model near the wall, thanks to the use of a blending function.
4.3 Mass transfer theory
The diusion process of some species from regions with higher concentrations to zones with lower concentration can be studied using two fundamental ap-proaches: the rst is the diusion theory based on the Fick's law. The second, which uses mass transfer coe cients, is an approximation that re-calls the concept of dierence of concentration, instead of gradient.
Considering the interface between two regions at dierent concentration val-ues, a ux can be evaluated as follows:
n = k(cinterf ace− cbulk) (4.14)
where:
nis the total ux across the interface [kg/m2s]. It includes both the diusive
and the convective transport of the interested specie;
cinterf ace and cbulk are the concentrations at the interface liquid-gas interface
and in the gas bulk;
k is the mass transfer coe cient [m/s]: it is the rate constant per unit of area of the movement of one species from the interface to the bulk.
The concept of mass transfer coe cient is often applied for prediction and explanation of experiments with transfer across some interfaces: the need to generalize the results for predicting other analogous phenomena has involved the use of correlations based on dimensionless numbers like the Reynolds and Schmidt numbers.
Mass transfer across interfaces
Let us consider a liquid-gas interface shown in the following picture:
The ux on the gas side is given by:
ng = kg,p(pg(bulk)− pg(interf ace)) (4.15)
while the ux on the liquid side is given in terms of the mole fractions: nliq = kliq,x(xliq(bulk)− xliq(interf ace)) (4.16)
Under equilibrium conditions, the validity of the Henry's law leads to express the interface gas pressure as KHxliq(interf ace) so that:
xliq(interf ace) = pg(interf ace) KH = ng kliq,x + xliq(bulk)
ng = kg,p(pg(bulk)− pg(interf ace))
(4.17) and nally it is possible to eliminate the values of pressure and mole fraction at the interface:
ng =
(pg(bulk)− KHxliq(bulk))
1/kg,p+ KH/kliq,x
(4.18) Through the electric analogy, it can be easily seen how the two terms at the denominator act as resistances to the passage of current due to the presence of the voltage at the nominator, the partial pressure dierence.
For the aims of this work, it could be useful to recall some correlations to express the mass transfer coe cients responsible of the transfer and transport of species in situation similar to the one implemented in ANSYS.
The Graetz-Nusselt problem
Let us consider a steady state laminar ow inside a tube of radius R with the inner surface able to exchange with the carrier gas.
For this kind of system the Sherwood number can be expressed as follows (See App.A for details):
Sh = 1.62 · Re1/3· Sc1/3· d
L 1/3
(4.19) where, being do the tube diameter, v the carrier gas velocity, DP o(Ar) the
polonium diusion coe cient in Argon and ν the Argon viscosity, Sh = k do
DP o(Ar)
is the Sherwood number, ratio between the convective and diusive mass transfer;
Re = do
v
ν is the Reynolds number, ratio between the inertial and viscous forces;
Sc = ν DP o(Ar)
is the Schmidt number, ratio between the momentum diusiv-ity (Argon viscosdiusiv-ity) and the mass diusivdiusiv-ity.
If we consider that:
kc = Sh
DP o(Ar)
do (4.20)
we obtain the expression for the mass transfer coe cient: kc= 1.62 DP o(Ar) do d2 ov LDP o(Ar) 1/3 (4.21) Mass transfer from a plate
Another example worthy of interest is the situation in which a carrier gas ows upon a at plate with releasing surface.
It is useful to recall the concept of laminar boundary layer together with the one of concentration boundary layer: the rst is the distance where it is located the 99% of the velocity variation, starting from zero on the wall. The second is the representative length of the penetration of the solute in the gas current.
Sherwood number can be expressed as (See App.B for details):
Sh = 0.646 · Re1/2· Sc1/3 (4.22)
Hence considering that:
kf = Sh
DP o(Ar)
L (4.23)
where L is the plate length, it is possible to obtain the expression for the mass transfer coe cient:
kf = 0.646
DP o(Ar)
L Re
1/2· Sc1/3 (4.24)
Mass transfer coe cient from Van Limpt's work
In the work of Van Limpt [38] the mass transfer during the transpiration experiment is evaluated making use of a geometrical approximation: the sample is no more on the surface of the tube but is considered as a concentric cylinder inside the outer one.
Under this assumption the averaged Sherwood number is evaluated according to: Shavg = 3 X i=1 Sh3i !1/3 (4.25) and: Sh1 = 3.66 + 1, 2 di do 0.5 Sh2 = 1.615 " 1 + 0, 14 di do 1/3# 3 r Re · Sc · dh L Sh3 = 2 1 + 22 · Sc 1/6 Re · Sc · d h L 0.5 (4.26) where:
di and do are the sample and tube diameter, respectively;
dh = di − do is the hydraulic diameter in the region of the tube where the
sample is located; Lis the sample length.
The determination of the correspondent mass transfer coe cient is ex-plained below:
kV anLimpt =
Shavg
dh
