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Nuclear Engineering and Technology 51 (2019) 394e401 Contents lists available at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article Sensitivity analysis of thermal-hydraulic parameters to study the corrosion intensity in nuclear power plant steam generators S. Tashakor a, *, A. Afsari b, M. Hashemi-Tilehnoee c a Department of Renewable Energy, Shiraz Branch, Islamic Azad University, Shiraz, Iran Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran c Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran b a r t i c l e i n f o a b s t r a c t Article history: Received 23 December 2017 Received in revised form 19 October 2018 Accepted 22 October 2018 Available online 2 November 2018 The failure of steam generators (SGs) due to corrosion is one of the most important problems in power plants. Impurities usually accumulate in the hot sides of SG and form deposits on the SG surfaces. In this paper, the sensitivity analysis of the accumulation of water impurities in the heat exchangers of nuclear power plants is presented. The convection-diffusion equation of the liquid phase on the heated surfaces is derived and then solved by the finite volume method. Also, the effects of the thermalehydraulic parameters in the form of dimensionless numbers, such as Peq, Peu, kp(relative solubility of impurity between the steam and water) on the impurities concentration are studied. © 2018 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Corrosion Steam generators Impurities Hideout and return 1. Introduction Steam generators (SGs) are key components in Pressurized Water Reactors (PWRs). Their reliability affects greatly the overall plant performance and availability. Worldwide experience shows that a significant number of operating PWRs have now corrosion or mechanical degradation problems in their SGs [1]. The impurities come from the secondary side systems which generally accumulate in the SG and form deposits on the tube sheet surfaces and tube support plates. The corrosion, which is caused by these deposits, has been identified as a major cause of heat transfer degradation in power plants, such that it can reduce the output power as low as 80% of full power [2]. Jahanfarnia and Tashakor developed a mathematical model for impurities hideout and return in a nuclear power plant steam generator. Hideout is a complex process which depends on the local geometry, thermo-hydraulic conditions and the solubility characteristics of dissolved impurities. The hideout and return phenomenon for dissolve impurities in the boiling flow is such that increasing heat flux and evaporation intensity make the impurities moving from bulk flow to the viscous sublayer and * Corresponding author. Tel.: þ98 917 1073160. E-mail address: saman.tashakor@yahoo.com (S. Tashakor). therefore the impurities concentration will be increased on heating surfaces [3e6]. Tashakor et al. [7] developed a numerical model to estimating the corrosion location in nuclear power plant steam generators. The results presented in their research show that average solubility coefficient of B2O3 in steam is more than solubility coefficient of SiO2 consequently the B2O3 concentration in the same heat flux is less than SiO2 average concentration in the viscose (laminar) sublayer. Research on the deposition model of particlelike corrosion product in SG tubes of PWR was carried out by Daogang Lu et al. [8]. The numerical results of the particles’ motion and deposition distribution in SG tubes showed good agreements with the experimental results. After the onset of boiling in the heated channel and also considering the solubility of impurities of water, which are much more than those of vapor, a part of impurities remain in the liquid phase due to evaporation in the viscous sublayer. The impurities concentrations increase in the liquid phase and only a small amount of impurities exit with the vapor. Convection and diffusion are the two phenomena, which have major effects on impurities concentrations changes. During the flow convection, impurities enter the viscous sublayer. By increasing their concentrations in the viscous sublayer, impurities move from the viscous sublayer to the bulk flow due to the diffusion phenomenon. Finally, in the equilibrium, impurities concentrations in the viscous sublayer reach the https://doi.org/10.1016/j.net.2018.10.021 1738-5733/© 2018 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 395 steady-state condition. Because of flow turbulence, there is no gradient of impurities concentrations out of the viscous sublayer [7]. Mass conservation equation will be written in viscous sublayer. The presented equations are solved with numerical methods. The sensitivity analysis will be presented based on the changes of thermal-hydraulic parameters in the form of dimensionless numbers. 2. Methods and materials The continuity equation for the liquid in the viscous sublayer is as follows [3,7,9,and10]: 
!  v
jð1  aÞrf þ V jð1  aÞ W rf þ jð1  aÞrf g ¼ 0 vt (1) The detailed nomenclatures for the variables and parameters are provided in the Nomenclature section. With assuming constant values for the void fraction (a), diffusion coefficient (D) and intensity of vaporization in the viscous sublayer (g), the velocity gradient in the channel can be obtained: ! V W ¼ g (2) As shown in Fig. 1, the distribution of impurities in the boiling flow is non-uniform only in the viscous sublayer. The impurities transfer takes place with three mechanisms; convection, diffusion (the negative sign indicates that the concentration gradient is negative), and outgoing vapor. The distribution of dissolved impurities in the viscous sublayer can be dealt with by the following convection-diffusion equation (impurities mass balance) [3,7,9,and10]: u ¼ Uux ! v ðjð1  aÞCÞ þ Vðjð1  aÞ W C  DVCÞ þ jð1  aÞkp:g:C vt |fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} where ux, wx the linear functions for axial and radial velocity. By considering a and j as constant values: convection Changes over time diffusion outgoing vapor ¼0 (3) where g is the intensity of evaporation, and kp is the relative solubility of impurity between the steam and water that given by: g¼ q gx ; kp jð1  aÞhfg rf d Impurity solubility coefficient in steam ¼ Impurity solubility coefficient in liquid (4) Fig. 1. The proposed convection and simple diffusion model. (7)  !  vC þ V W C  DVC þ gkp C ¼ 0 vt (8) Substituting Eq. (2) in (8) yields:  !  ! vC þ V W C  DVC  V W kp C ¼ 0 vt (9)  !  ! vC þ V W C  VðDVCÞ  V W kp C ¼ 0 vt (10)  !   !  ! V W C ¼ W VC þ CV W (11)  ! vC ! þ W VC  VðDVCÞ þ V W 1  Kp C ¼ 0 vt (12) where, gx is dimensionless distribution function: ðd g dx ¼ 1 (5) By considering steady state conditions, Eq. (13) and Eq. (14) is obtained. 0 gx ¼ vwx ; wx ¼ vx
!  ! WC  VðDVCÞ þ V W 1  kp C ¼ 0 ðx gx dx 0 v2 C v2 C þ D vx2 vz2 The radial and axial velocity profile are defined as: ðx W¼ gdx ¼ 0 ðx q g dx jð1  aÞhfg rf d x 0 (6) þ q ! þ q vwx  jð1  aÞhfg rf d vx ¼0 vC jð1  aÞhfg rf d vx  Uðux Þ (13) vC vz  1  kp C (14) 396 S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 The above equation can be rewritten in dimensionless form as below (see Refs. [3,10]): The applied boundary conditions are:  vC 1 v vC þ D vt Peq vx vx þ wx  vC Peu vC vwx   u þ 1  kp C ¼ 0 vx Peq x vz vx (15) where [3,7,10]: vC ¼0 vx x¼0 (16) Cz¼0 ¼ 1 (17) x ¼ xd0 < x < 1wx ¼ wx d; wx ¼ x x Z ¼ zH 0 < z < 1ux ¼ ux ; ux ¼ 2 qd Peq ¼ jð1  aÞ hfg rf D 2 ; Ud Peu ¼ DH Gv Impurity solubility coefficient in steam ¼ ; kp ¼ Impurity solubility coefficient in liquid Gin Cin !n 1 Fig. 2 illustrates a simple model like to the second cycle of SG in PWR plants to study the impurities hideout and return. The relevant mechanism contains three basic parts; Evaporation channel, Water and steam separator, connection tubes. It is supposed that all the water entering the evaporation channel evaporates. The feed water with mass flow rate Gin enters the separator. A low percent of mass flow rate, Gbd, exit from the separator with impurity CP by the blowdown output channel and, finally the steam leave the separator with mass flow rate GV. The mass balance of impurities is written as follows. Gin ¼ Gv þ Gb:d  GF þ GF ¼ ð1 þ PÞGv Cp Gb.d Cp rg rf Gv ¼ Separator (18) qS hfg (19) Impurities hideout and return in the evaporation channel is affected by diffusion and convection. The radial velocity of the convection at the boundary of viscous sublayer is as follows: Wd ¼ CB q jðl  aÞhfg rf d d ¼ Peq D (20) d Impurities mass balance equation in the evaporator is as Table 1 Thermal hydraulic and design values. Gf Cp Evaporation channel Parameters Values Diffusion coefficient ðDÞ Viscous sublayer thickness (d) Height of evaporator channel ðHÞ Average axial velocity of fluid ðUÞ SG secondary side pressure ðPÞ Void fraction ðaÞ 109 (m2 s1) 0.0001 (m) 1 (m) 0.1 (m/s) 7 (MPa) 0.5 0.01 4.0 1.9 l Fig. 2. Simple hideout and return model considered in the calculations. k n ðSiO2 Þ S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 397 ! vCB q  CB S vx jð1  aÞrf hfg |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} D changes in the impurities concentration of the bulk fluid due to the diffusion and convection . ðGF CP  ðGF  GV ÞCB Þ rf |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} þ Changes in the impurities concentration ¼ of the fluid passing through the evaporator channel vC V0 B vtffl} |fflfflffl{zfflffl Changes (21) over time (a) (b) (c) (d) Fig. 3. The variation of SiO2 distribution in viscous sublayer in different Peq number. (a: Peq ¼ 3.17, b: Peq ¼ 6.34, C: Peq ¼ 12.68, d: Peq ¼ 25.36). 398 S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 K¼ GF GV Dimensionless form of impurities mass balance in viscous sublayer boundary is as follows: (22) ! . vCB q  C S þ KGV CP  ðKGV  GV ÞCB Þ rf D vx jð1  aÞrf hfg B vC qd qd C þ ðKCP  ðK  1ÞCB Þ ÞS   vx x¼1 jð1  aÞhfg rf D B rf hfg D 2 ¼ vC ¼ V0 B vt (23) V0 d vCB Dd vt (25) ! vC qd qd C þ KCP  ðK  1ÞCB   S vx x¼1 jð1  aÞhfg rf D B rf hfg D Vo vCB ¼ d vFo Upon substitution of Eq. (22) into Eq. (23) yields: ! vCB q q vC  C þ KCP  ðK  1ÞCB D S ¼ V0 B vx jð1  aÞrf hfg B rf hfg vt (24) (a) (b) (c) Fig. 4. The variation of SiO2 distribution in viscous sublayer in different peu. (a: Peu ¼ 1, b: Peu ¼ 2, c: Peu ¼ 3). (26) S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 down to repair or blades replacement. vC Vo vCB   Peq CB þ ðKCp  ðK  1ÞCB Peq ¼ vx x¼1 d:S vFo (27) The equation of impurities mass conservation in the evaporator channel, which can be used as the boundary condition for the viscous sublayer is obtained as follows: K¼ Gf Gv p¼ 1þl Gb:d ky ¼ l Gv 1þ k Fo ¼ D d2 t l¼ Gb:d Gin U¼ Vo d:S   vC vC þ ðCP  CB ÞK Peq ¼ U B  vx x¼1 vFo (28) Gin Cin þ ðGF  Gv ÞCB ¼ ðGF þ Gb:d ÞCP (29) CP ¼ ð1 þ PÞCin þ ðK  1ÞCB PþK (30)   vC ð1 þ pÞCin þ ðK  1ÞCB vC  CB ÞK Peq  ¼U B vx x¼1 Pþk vFo (31) vC C  CB vC Kð1 þ pÞPeq ¼ U B þ in  vx x¼1 PþK vFo (32)   Pþ1 vC vC KPeq C þ U þ Pþk vFo vx ky Peq C þ U vC vC þ vFo vx x¼1 x¼1  ðp þ 1ÞK Peq Cin pþK ¼ ky Peq Cin 3.1. Sensitivity analysis of SiO2 distribution in the different Peq number The SiO2 density changes in different Peq, normalized with the inlet impurity concentration, are presented in this section (Fig. 3). In this case, total flow pressure and flow velocity are assumed fix. In the conditions of increasing heat flux from low values up to 300 kW which lead to increasing Peq, the hideout of impurities in viscous sublayer is studied (the heat flux of SG tube surface for the typical PWR is ranged from 120 to 350 kW/m2). Changes in colors in the latter figures demonstrate the impurity concentration gradient. The red colors indicate impurities accumulation region in the viscous sublayer. Increasing Peq and evaporation intensity leads to the impurities move from bulk flow to the viscous sublayer so the impurity concentration will increase in that region. On the other hand, the molecular diffusion phenomenon changes the impurities concentration in the opposite direction gradually. By decreasing the Peq, as a consequence of diffusion, the concentrated impurities transfer to the bulk stream and the impurities concentration increase in that section [4e7] (see Fig. 4). 3.2. Sensitivity analysis of SiO2 distribution in different Peu number  ¼ 399 (33) (34) Where, C is the concentration of dissolved impurities in the unit of volume (kg/m3) and kp is the coefficient of relative solubility of impurity between vapor and water. S is the surface area of viscous sublayer. d is the viscous sublayer thickness and Vo is the mainstream volume in the evaporator, Gb:d is the blowdown flow rate, Gin is the feed water flow rate with initial concentration Cin, Gf is the entered flow rate to evaporator channel, and Gv is the exited vapor rate from evaporator channel. The dimensionless differential equations representing convection and diffusion are solved numerically by finite volume method upwind scheme [11,12]. Sensitivity analysis of thermal-hydraulic parameters, in the form of dimensionless numbers under the steady-state condition, is done with the code written in FORTRAN and the results are plotted with Tecplot software. Thermal-hydraulic design values are shown in Table 1 [3,6,and13]. In this section, the SiO2 density changes for various Peu (which is normalized with the inlet impurity) is discussed (Fig. 3). Here, total flow pressure and heat flux are assumed constant. The minimum recommended liquid velocity inside PWR SGs tubes is 1.0 m/s, while the maximum is 2.5e3.0 m/s [14]. With increasing the Peu, convection will be increased along the channel and the impurities will move from the laminar (or viscous) sublayer and heating surfaces to the bulk flow very fast. 3.3. Sensitivity analysis of SiO2 distribution in different kp number In this section, the SiO2 density changes for various kp (which is normalized with the inlet impurity) is discussed (Fig. 5). Here, the velocity of flow and the heat flux are assumed constant. Loss of 3. Result and discussion Contaminated impurities in PWR secondary side play an important role in the hideout and return in SG crevice and localized corrosion. Especially, colloidal SiO2 affects the hideout and return, because colloidal SiO2 has low solubility in the high-temperature water such as PWR secondary coolant. Among many contaminants in the steam/water circuit, SiO2 plays a significant role in process monitoring, because it is highly soluble in steam and extremely difficult to remove it from steam/water. SiO2 generates a coating on surfaces which is very difficult to be removed (even with acid) and causes a loss in thermal efficiency. The steam, when passing through the turbine, comes into contact with the turbine blades and is cooled down. Hence the silica dissolved in the steam deposits on the blades and in the worst case, the plant must be shut Fig. 5. The variation of kp (relative solubility of impurity between the steam and water) in different pressure. 400 S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 occurrence of corrosion is observed in the upper and outlet parts of the evaporation channel (red areas in the figures) (the intensity of corrosion depends on heat flux and impurity concentration close to the heating surface). Nesta and Bennett [15] and Kumar Gautam [16] shown that fouling rates decrease with increasing flow velocity where these observations are consistent with the results of this research. feedwater in the secondary coolant system leads to a loss-of-heat sink accident (LOHA). With decreasing feedwater flow rate, in constant heat flux condition, the vaporization rate increases, considering that solubility of impurity in the steam is less than liquid, the density of impurities near the SG heating surface will be increased during loss-of-heat sink accident of the system. On the other hand, with decreasing system pressure, kp decreases (Fig. 5). As a result, the ratio of the impurity solubility in the vapor to the liquid decreases, which leads to an increase in the concentration of impurities in the viscous sublayer (secondary water in the PWR steam generator boils at a pressure of approximately 7 MPa, which corresponds to temperature 260  C). The results of this simulation are consistent with those of Odar in 2004 [2]. In both studies, the accumulation of impurities and the 4. Conclusion The changes of impurities concentration close to the heated surface (viscous sublayer) are investigated relative to thermalhydraulic parameters, in the form of dimensionless numbers under the steady-state condition. The convection-diffusion equations (a) (b) (c) Fig. 6. The variation of SiO2 distribution in viscous sublayer in different kp. (a: kp ¼ 0.002,b: kp ¼ 0.0015, c: kp ¼ 0.001). S. Tashakor et al. / Nuclear Engineering and Technology 51 (2019) 394e401 are solved by finite volume numerical method. The results show that as Peq (heat flux) increases, the impurities concentration in the laminar sublayer will be increased and hence, the intensity of corrosion increases. Also with increasing Peu (the flow velocity) convection and transferring impurities from heated surfaces to the bulk flow will be increased and the impurities density and the intensity of corrosion near heated surface decreases. By decreasing the SG feedwater rate or loss of heat sink accident (LOHA), the intensity of evaporation is increased also the relative solubility of impurity between the steam and water is decreased hence, the corrosion increases. Also, the reverse of the above results has been confirmed. The results show that the most concentration of impurities is observed in the heated channel output, so the most corrosion rate occurs in this area. Finally, in higher pressure and velocity of flow and lower heat flux, fewer deposits of soluble impurities can be seen. References [1] EPRI Steam Generator Progress Report, 2000. Rev 15. [2] S. Odar, Water chemistry measurement to improve steam generator performance, in: 14th International Conferences on the Properties of Water and Steam in Kyoto, 2004, pp. 531e538. [3] G.R. Jahanfarnia, S. Tashakor, Mathematical model of impurities hideout and return in nuclear power plant steam generator, Prog. Nucl. Energy 51 (2009) 680e685. [4] V.V. Yagov, Heat transfer with developed nucleate boiling of liquid, Therm. Eng. 35 (1998) 65. [5] Y. He, M. Shoji, S. Maruyama, Numerical study of high heat flux pool boiling heat transfer, Int. J. Heat Mass Tran. 44 (2001) 2357. [6] T. Kumada, H. Sakashita, Pool boiling heat transfer-II thickness of liquid macro layer formed beneath vapor masses, Int. J. Heat Mass Tran. 38 (6) (1995) 979e987. [7] S. Tashakor, G. Jahanfarnia, A. Kebriaee, Numerical model for estimation of corrosion location in nuclear power plant steam generator, Nucl. Eng. Des. 241 (2011) 95e100. [8] Daogang Lu, Bo Yuan, Licun Wu, Zhongying Ma, Yixue Chen, Research on the deposition model of particle-like corrosion product in SG tubes of PWR, Ann. Nucl. Energy 81 (2015) 98e105. [9] J. Bear, Y. Bachmat, Macroscopic modeling of transport phenomena in porous media: applications to mass momentum and energy transfer, Transport Porous Media 1 (1986) 241e269. [10] G.R. Jahanfarnia, V.I. Gorburov, Estimation of Viscous Sublayer Thickness and Distribution of Impurities in Nucleate Boiling. Monte Carlo 2005, 2005. Chattanooga, Tennessee, USA, 17e21 April. [11] A. Anderson, J.C. Tanehill, Computational Flow Mechanics and Heat Transfer, McGraw-Hill, New York, 1984. [12] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Flow Dynamics the Finite Volume Method, John Wiley Inc, 1995, pp. 100e190. [13] D.H. Charlesworth, The deposition of corrosion products in boiling water systems, Chem. Eng. Prog. Symp. Ser. 661 (104) (1970) 21. [14] R. Mukherjee, Effectively Design shell-and etube Heat exchangers, Chem. Eng. Prog. (1998) 21. Feb.. 54. 401 [15] J. Nesta, C.A. Bennett, Fouling mitigation by design, ECI symposium series, in: Hans Müller-Steinhagen, M. Reza Malayeri, A. Paul Watkinson (Eds.), in: . 54. (Ed.), Proceedings of 6th International Conference on Heat Exchanger Fouling and Cleaning -Challenges and Opportunities, Volume RP2, 2005. Engineering Conferences International, Kloster Irsee, Germany. [16] R. Kumar Gautam, Fouling in Process Apparatuses, Research Master Thesis, Czech Technical University in Prague Faculty of Mechanical Engineering, Department of Process Engineering, 2018. Available: https://dspace.cvut.cz/ handle/10467/74090. Nomenclature C: Concentration of dissolve impurities (kg/m3) CB: Impurity concentration in evaporator (kg/m3) Cin: Initial concentration (kg/m3) Cp: Blowdown impurity concentration (kg/m3) D: Diffusion coefficient (m2/s) FO: Fourier number g: Intensity of vaporization in the viscous sublayer gx: Dimensionless distribution function Gb.d: Blow down flow rate (kg/s) Gf: Entered flow rate to evaporator channel (kg/s) Gin: Feed water flow rate (kg/s) Gv: Exited vapor rate to the evaporator channel (kg/s) H: Height of evaporator channel(m) hfg: Enthalpy of evaporation (kJ/kg) kP: Coefficient of impurity relative solubility between the steam and water n: Ratio of impurity solubility in the vapor to liquid P: pressure in the secondary side of the steam generator (MPa) Pe: Peclet number q: Surface Heat flux (kW/m2) S: Surface area of the viscous sublayer(m2) t: Time per second (s) U: Average axial velocity of flow (m/s) u: Axial velocity profile ux,ux: Linear function for axial velocity (m/s) V: Main stream volume(m/s) ! W : Vector of liquid motion velocity (m/s) W: Radial velocity of flow(m/s) wx,wx: Linear function for radial velocity (m/s) x, z: width and length of viscous sublayer Greek symbols rf, rg: Liquid, and vapor density d: Viscous sublayer thickness a: void fraction j: Coefficient of channel wall roughness Subscript b.d: Blowdown flow rate F: Flow V: Vapor x, z: Be a sign of width and length of viscous sublayer in the dimensionless form