Homogeneous bubble nucleation limit of lead

Liquid heavy metal coolant in a fast reactor, as well as in accelerator driven systems, is exhibited to large thermal and pressure shocks which can cause cavitation in the coolant. Here we calculated the work of the critical bubble formation in the lead coolant and the nucleation rate in terms of th...

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Veröffentlicht in:	Вопросы атомной науки и техники
Datum:	2012
Hauptverfasser:	Abyzov, A.S., Schmelzer, J.W.P., Davydov, L.N., Slezov, V.V.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Schlagworte:	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
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spelling	Abyzov, A.S. Schmelzer, J.W.P. Davydov, L.N. Slezov, V.V. 2016-10-14T10:37:43Z 2016-10-14T10:37:43Z 2012 Homogeneous bubble nucleation limit of lead / A.S. Abyzov, J.W.P. Schmelzer, L.N. Davydov, V.V. Slezov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 283-287. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 64.60.Qb, 72.15.Cz https://nasplib.isofts.kiev.ua/handle/123456789/107158 Liquid heavy metal coolant in a fast reactor, as well as in accelerator driven systems, is exhibited to large thermal and pressure shocks which can cause cavitation in the coolant. Here we calculated the work of the critical bubble formation in the lead coolant and the nucleation rate in terms of the generalized Gibbs approach. It is demonstrated that such approach provides a more adequate description of the process of bubble nucleation as compared with the classical nucleation theory. Теплоносители на основе жидких тяжелых металлов в быстрых реакторах и реакторах, управляемых ускорителем, подвержены значительным тепловым и гидравлическим ударам, что может приводить к кавитации теплоносителя. Рассчитаны работа образования критических пузырьков в свинцовом теплоносителе и скорость их зарождения в рамках обобщенного подхода Гиббса. Показано, что такой подход обеспечивает более адекватное описание процесса зарождения пузырьков по сравнению с классической теорией нуклеации. Теплоносії на основі рідких важких металів у швидких реакторах і реакторах, керованих прискорювачем, схильні до значних теплових і гідравлічних ударів, що може призводити до кавітації теплоносія. Розраховані робота утворення критичних пухирців в свинцевому теплоносії та швидкість їх зародження в рамках узагальненого підходу Гіббса. Показано, що такий підхід забезпечує більш адекватний опис процесу зародження пухирців в порівнянні з класичною теорією нуклеації. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Homogeneous bubble nucleation limit of lead Порог гомогенного зарождения пузырьков в свинце Поріг гомогенного зародження пухирців у свинці Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Homogeneous bubble nucleation limit of lead
spellingShingle	Homogeneous bubble nucleation limit of lead Abyzov, A.S. Schmelzer, J.W.P. Davydov, L.N. Slezov, V.V. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
title_short	Homogeneous bubble nucleation limit of lead
title_full	Homogeneous bubble nucleation limit of lead
title_fullStr	Homogeneous bubble nucleation limit of lead
title_full_unstemmed	Homogeneous bubble nucleation limit of lead
title_sort	homogeneous bubble nucleation limit of lead
author	Abyzov, A.S. Schmelzer, J.W.P. Davydov, L.N. Slezov, V.V.
author_facet	Abyzov, A.S. Schmelzer, J.W.P. Davydov, L.N. Slezov, V.V.
topic	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
topic_facet	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
publishDate	2012
language	English
container_title	Вопросы атомной науки и техники
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format	Article
title_alt	Порог гомогенного зарождения пузырьков в свинце Поріг гомогенного зародження пухирців у свинці
description	Liquid heavy metal coolant in a fast reactor, as well as in accelerator driven systems, is exhibited to large thermal and pressure shocks which can cause cavitation in the coolant. Here we calculated the work of the critical bubble formation in the lead coolant and the nucleation rate in terms of the generalized Gibbs approach. It is demonstrated that such approach provides a more adequate description of the process of bubble nucleation as compared with the classical nucleation theory. Теплоносители на основе жидких тяжелых металлов в быстрых реакторах и реакторах, управляемых ускорителем, подвержены значительным тепловым и гидравлическим ударам, что может приводить к кавитации теплоносителя. Рассчитаны работа образования критических пузырьков в свинцовом теплоносителе и скорость их зарождения в рамках обобщенного подхода Гиббса. Показано, что такой подход обеспечивает более адекватное описание процесса зарождения пузырьков по сравнению с классической теорией нуклеации. Теплоносії на основі рідких важких металів у швидких реакторах і реакторах, керованих прискорювачем, схильні до значних теплових і гідравлічних ударів, що може призводити до кавітації теплоносія. Розраховані робота утворення критичних пухирців в свинцевому теплоносії та швидкість їх зародження в рамках узагальненого підходу Гіббса. Показано, що такий підхід забезпечує більш адекватний опис процесу зародження пухирців в порівнянні з класичною теорією нуклеації.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/107158
citation_txt	Homogeneous bubble nucleation limit of lead / A.S. Abyzov, J.W.P. Schmelzer, L.N. Davydov, V.V. Slezov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 283-287. — Бібліогр.: 15 назв. — англ.
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last_indexed	2025-11-26T18:52:01Z
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fulltext	HOMOGENEOUS BUBBLE NUCLEATION LIMIT OF LEAD A.S. Abyzov 1∗, J.W.P. Schmelzer 2, L.N. Davydov 1, and V.V. Slezov 1 1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine 2Institut für Physik der Universität Rostock, 18051, Rostock, Germany (Received October 31, 2011) Liquid heavy metal coolant in a fast reactor, as well as in accelerator driven systems, is exhibited to large thermal and pressure shocks which can cause cavitation in the coolant. Here we calculated the work of the critical bubble formation in the lead coolant and the nucleation rate in terms of the generalized Gibbs approach. It is demonstrated that such approach provides a more adequate description of the process of bubble nucleation as compared with the classical nucleation theory. PACS: 64.60.Qb, 72.15.Cz 1. INTRODUCTION Liquid heavy metal heat-carrier (lead, bismuth, eu- tectic Pb-Bi alloy, mercury) in fast reactors and accelerator-driven reactors subject to significant ther- mal and hydraulic shocks. Increase in temperature and lowering of pressure (down to negative values at negative phase of a pressure wave period) can result in a cavitation in the coolant. The newly formed bub- bles filled with metal vapor, together with a coolant circulation can come to the high pressure and low temperature regions, where they collapse and dam- age the construction materials. There are methods of controlling pressure shocks, however, temperature shocks cannot be avoided. In this paper we determine the criterions of homogeneous bubble nucleation in the liquid metal lead heat-carrier using the modified Gibbs approach [1–3] and compare the results with the classical nucleation theory (CNT). We approached the problem in three steps. In the first step (Sec. 2), we calculated the phase equilib- rium properties, the stability limit and various other properties of lead by using a slightly modified version of the equation of state [2] proposed by Redlich and Kwong [4] (see also [5–7]). In the next step (Sec. 3), we calculated the work of critical bubble formation in lead as well as the rate of homogeneous nucleation in the pressure-temperature range defined according to the results of the previous section. The paper is com- pleted by a short summary and discussion (Sec. 4). 2. MODEL SYSTEM For the description of lead (Pb) in both the liquid and gas phases, we will apply a slightly modified Redlich- Kwong-type equation of state as proposed by Morita et al. (see [4, 5] and, in particular, Eq. (1) in [2]). It reads p = RT M(v − b) − a(T ) v(v + c) , (1) a(T ) = ac (T/Tc) m , (2) where R = 8.314 J/mol is the universal gas con- stant, p is pressure, v is molar volume, T is tempera- ture, ac, b, c and m are the model parameters specific for the substance, Tc is critical temperature. We employ further dimensionless variables Π ≡ p pc , ω ≡ v vc , θ ≡ T Tc , (3) where vc is the molar volume, pc the pressure both at the critical point with the critical temperature, Tc. These parameters can be determined from equation (1) in the common way via ∂p ∂v ∣∣∣∣ T = ∂2p ∂v2 ∣∣∣∣ T = 0 at T = Tc. (4) The equation of state in reduced variables is given by Π(θ, ω) = θ χc(ω − β) − α(θ) ω(ω + ξ) . (5) Here χc = pcvc RTc (6) is the reduced critical compressibility, and α(θ) = acθ m pcv2 c = αθm, (7) β = b/vc, ξ = c/vc. (8) According to [6] we have then α = 5.9, β = 1.7 · 10−3, ξ = 0.768, m = −0.01. (9) From equations (1) and (4) we get pc = 121 MPa, ρc = 1920 kg/m3 , vc = 5.2 · 10−4 m3/kg, Tc = 5843 K. (10) ∗Corresponding author E-mail address: abyzov@kipt.kharkov.ua PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 283-287. 283 The location of the classical spinodal curve can be found by the determination of the extrema of the thermal equation of state, Π(θ, ω) (5), considering the temperature θ as constant. By taking the deriva- tive of Π(θ, ω) with respect to temperature, we obtain from equation (5) the result ∂ ∂ω Π(θ, ω) = α(θ)(2ω + ξ) ω2(ω + ξ)2 − θ χc(ω − β)2 = 0. (11) For θ < 1 , this equation has two positive solutions ω (left) sp and ω (right) sp for ω corresponding to the spe- cific volumes of the both macrophases at the spinodal curves (or at the limits of metastability). Similarly, the binodal curves give the values of the specific vol- umes of the liquid and the gas phases coexisting in thermal equilibrium at a planar interface. From the left branch of the binodal curve, we get the specific volume of the liquid phase ( ω(eq) l = ω (left) b ), from the right branch of the binodal curve, we obtain the specific volume of the gas ( ω(eq) g = ω (right) b ). For θ = 1 , both solutions coincide in the critical point ( ω(eq) l = ω (eq) g = ωc = 1 ), again. Consequently, in order to determine the specific volumes of the liquid and the gas at some given temperature in the range θ ≤ 1 , we have to specify the location of the bin- odal curve. The location of the binodal curve may be determined from the necessary thermodynamic equi- librium conditions (for planar interfaces) – equality of pressure and chemical potentials – via the solution of the set of equations Πl (ωl, θ) = Πg(ωg, θ), μl(ωl, θ) = μg(ωg, θ). (12) Here by μ the chemical potential of the atoms or molecules in the liquid (l) and the gas (g) are de- noted. Having the equation for the reduced pres- sure (cf. Eq. (5)), we have now to determine in ad- dition the chemical potential in dependence on pres- sure and temperature (see Sec. 2). Isotherms for lead (5) for different values of the reduced temperature θ = 0.4, 0.65, 0.8, 0.891 and 0.92 are shown in Fig. 1, dashed and dashed-dotted curves present the binodal and spinodal curves, correspondingly. For isothermal processes, the change of the Helmholtz free energy, dF , may be expressed as dF = −pdV + μdn. (13) Here V is the volume of the system and n the number of moles in it. For a given fixed mole number, n, of the substance (n = const.), we have dφn = −pdv, φn = F n , v = V n , (14) or, in reduced variables, d ( φn pcvc ) = −Πdω . (15) Reduced volume, � R ed u ce d p re ss u re , � 0.1 1 10 100 1.25� 1� 0.75� 0.5� 0.25� 0 0.25 0.5 0.75 1 1.25 1.5 � �� m in = sp in o d albi no da l �� Fig. 1. Isotherms of mercury as described via equation (5) for different values of the reduced temperature, from θ = 0.4 (bottom curve) to θ = 1 (upper curve) Combining equation (15) with the equation of the state, equation (5), we obtain φn pcvc = − [ α(θ) ξ ln ( ξ ω + 1 ) + θ · ln(ω − β) χc ] . (16) Alternatively, the isochoric change of the Helmholtz free energy is given at constant temperature by dF = μdn. (17) From equation (17), dφv = − μ v2 dv, φv = F V . (18) On the other side, the functions dφv and φn are con- nected by F = φnn = φvV, φv = φn v . (19) With equation (16), we have then φv = pc ω [ α(θ) ξ ln ( ξ ω + 1 ) + θ · ln(ω − β) χc ] . (20) With equations (18) and (20), the expression for the chemical potential of a heavy liquid metal (HLM) can be obtained then via μ = −v2 ∂φv ∂v = −veω 2 ∂φv ∂v . (21) This relation yields μ(ω,θ) pcvc = − [ α(θ) ω+ξ + θ·ω χc(β−ω)+ + α(θ) ξ ln ( ξ ω + 1 ) + θ·ln(ω−β) χc ] + ψ(θ) . (22) In addition to the bulk properties of the system, we have to know the value of the surface tension, σ, for planar interfaces in dependence on the parameters describing the state of both phases. The following form was chosen for our calculation [3, 8–10] σ(ωg, ωl, θ) = Θ(θ) [ 1 ωl − 1 ωg ]δ , (23) 284 where Θ(θ) = A [ 1 ω (left) b − 1 ω (right) b ]k−δ , δ = 2, (24) and A and k are constant parameters. Comparison of equations (23) and (24) with experimental data [11, 12] σ(T ) = 0.4857− 0.000066 · T (25) (valid in this form only for temperatures far below the critical temperature; here the temperature is given in Kelvin and the surface tension in J/m2) at ωl = ω (left) b and ωg = ω (right) b yields A = 40.5 · 10−3 J/m2 , k = 1.4. (26) 3. DETERMINATION OF THE WORK OF CRITICAL CLUSTER FORMATION Let us assume that the system is brought suddenly into a metastable state located between binodal curve and spinodal curve on the liquid side. Then, by nu- cleation and growth processes, bubbles may appear spontaneously in the liquid and a phase separation takes place [13]. Based on the relations outlined above, we will find the temperature and pressure dependence of the parameters of the critical vapour cluster formation. We start with the general expres- sion for the change of the thermodynamic potential ΔG = σA+ (p− pα)Vα + ∑ j njα(μjα − μjβ) . (27) Here the subscript α specifies the parameters of the cluster (bubble) phase while β refers to the ambient liquid phase. This relation holds as long as the state of the liquid remains unchanged by the formation of one bubble. For a one-component system, this ex- pression is reduced to ΔG = σA+ (p− pα)Vα + nα(μα − μβ) . (28) As independent variables, we selected the radius of the bubble, r, and the molar volume of the gas phase in the bubble. Similarly to [2, 3, 14] we arrive then at Δg(r, ωg, ωl, θ) kBT = 3 ( 1 ωl − 1 ωg )δ r2+2f(ωg, ωl, θ)r3 , (29) where the following notations have been introduced: f(ωg, ωl, θ) = Π(ωg, θ) − Π(ωl, θ)+ + (ωgpcvc) −1 (μ(ωl, θ) − μ(ωg, θ)) , (30) g ≡ G Ω1 , Ω1 = 16π 3 1 p2 ckBTcθ Θ3(θ) , (31) r ≡ R Rσ , Rσ = 2 pc Θ(θ). (32) The Gibbs free energy surface for the metastable initial state has a typical saddle shape near to the configuration corresponding to a bubble of critical size (see Fig. 2, θ = 0.75, Ω1 = 0.44) in the space of critical radius-molar volume and work of cluster formation. W o rk o f cl u st er f o rm at io n , � G k T / B 2 4 6 Bubble density, , kg/m � g 3 0 1000 300 100 Rad iu s o f b ubble, n m 2.0 8.0 0 Saddle point, = 3.91 = 1.015 nm = 920 kg/m � � G k T R c/ B c 3 3000 6.0 4.0 Fig. 2. Gibbs free energy surface for metastable initial state, θ = 0.75, Ω1 = 0.44 The critical point position is determined by the following set of equations ∂Δg(r, ωg, ωl, θ) ∂r = 0, ∂Δg(r, ωg, ωl, θ) ∂ωg = 0. (33) The dependence of the critical cluster parameters on the negative pressure, p, are shown in Figures 3- 5, for θ = 0.70. The positions of the binodal and spinodal curves are given then by ω (left) b = 0.311 , pb = −0.254 , and ω (left) sp = 0.44 , psp = 1.174 , cor- respondingly. psp = 1.812 . Compare the obtained results with the classical nucleation theory (CNT). The work of formation of critical nucleus ΔGclass is determined in this case by the following equation (see, for example, [13]) ΔGclass kBT = 16π 3 σ3 (p− pg)2kBT . (34) Fig. 3 presents the dependence of gas density in critical nucleus on the density of liquid. Close to bin- odal (at very small metastability) the density of gas in a bubble only insignificantly differs from its equi- librium value (the same as in CNT). However, as the metastability increases the gas density in a bubble grows, coming closer to the density of the surround- ing liquid taken at spinodal. At the same time (see Eq. (23)) the bubble surface energy decreases, and, correspondingly, the work of nucleus formation also decreases. The dependence of the work of critical nucleus for- mation ΔGc/kBT , on negative pressure in a general- ized Gibbs model (blue line) and CNT (green line) is shown in Fig. 4. It is evident that in CNT the work 285 of critical nucleus formation does not “feel” spinodal, while for in the generalized Gibbs model the work decreases with the approach to spinodal. Indeed, at spinodal the gas density in a bubble coincides with the density of the liquid (see Fig. 3). It means, that such “nucleus” does not differ from a surrounding it medium and, naturally, the work of its formation is equal to zero. Therefore, the nucleation process close to spinodal goes not through a saddle point, but through a ridge of the potential ΔG (in details see [15]). The behaviour of the critical radius with the ap- proach to the binodal also differs in these theories: in the generalized Gibbs model the critical radius sharply increases (however, this does not influence the nucleation: as was mentioned above, the nucle- ation process close to spinodal does not pass through the saddle point), and in CNT there is no singularity (Fig. 5). The nucleation rate is determined by the following equation I = I0 exp ( −ΔGc kBT ) . (35) 2 2.3 2.6 2.9 3.2 3.5 0.1 1 10 Liquid density, /� �l c B u b b le g as d en si ty , / � � g c �sp � � �c sp (left) / � � c b (right)/S pi no da l, � sp � � � c sp(l ef t) / B in od al , � b � � � c b(l ef t) / Fig. 3. Dependence of gas density in the bubble on liquid density C ri ti ca l w o rk , � G k T c / B 0.5� 0 0.5 1 1.5 2 1 10 100 10 3 10 4 10 5 Pressure, /p pc generalized Gibbs CNT B in o d al S p in o d al Fig. 4. Dependence of work of the critical bubble formation, ΔGc/kBT , on negative pressure C ri ti ca l ra d iu s, n m �� 0.1 1 10 100 10 3 B in o d al Pressure, /p pc � � � � � � � CNT generalized Gibbs S p in o d al Fig. 5. Dependence of the radius of the critical bubble, Rc, on negative pressure Fig. 6 presents the dependence of the nucleation rate on negative pressure in the generalized Gibbs model (blue) and CNT (green) (in calculation the preexponential factor equaled I0 ≈ 1041 m−3 s−1). It is evident that the nucleation rate calculated in the classical theory is several orders less than the gener- alized Gibbs model predicts. S p in o d al CNT Pressure, /p pc N u cl ea ti o n r at e, , m s J -3 -1 � � � � � � 10 10� 10 5� 1 105 1010 10 15 10 20 10 25 1030 1035 1040 ge ne ra liz ed Gibbs Fig. 6. Dependence of nucleation rate J on nega- tive pressure 4. CONCLUSIONS Near the binodal the generalized Gibbs approach leads to the same results as the classical nucleation theory, but noticeably different ones for enough large degree of metastability, that is close to the spinodal (see Figs. 7, 10 and [1, 3, 14]). Therefore, the con- sidered approach is of importance for cavitational processes, which take place mainly at large metasta- bility. It was found that the nucleation rate, calcu- lated within the generalized Gibbs model is several orders higher than the classical theory predicts, that is the limit of cavitational stability is noticeably lower than in the classical theory. Note that the calcu- lations were done only for the case of homogeneous bubble nucleation in the coolant bulk. However, the heterogeneous nucleation at impurities in the coolant 286 or on the pipelines walls is also possible. The work of formation of critical bubbles is in this case lower, but also the number of nucleation sites is much less, there- fore the influence of heterogeneous nucleation on the threshold of cavitational stability requires additional investigation. References 1. V.V. Slezov. Kinetics of First-Order Phase Tran- sitions. Weinheim: “Wiley-VCH Publishers”, 2009, 429 p. 2. A.R. Imre, A.S. Abyzov, I.F. Barna, and J.W.P. Schmelzer. Homogeneous bubble nucleation limit of mercury under the normal working condi- tions of the planned European spallation neutron source // Eur. Phys. J. 2011, v. B79, p. 107-113. 3. J.W.P. Schmelzer, J. Schmelzer, Jr. Kinetics of bubble formation and the tensile strength of liquids // Atmospheric Research. 2003, v. 65, p. 303-324. 4. O. Redlich, J.N.S. Kwong. On the Thermody- namics of Solutions. V. An Equation of State. 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Homogeneous bubble nucleation limit of lead

Institution

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