Numerical Study on Subcooled Pool Boiling

Transcription

1 Progress in NUCLEAR SCIENCE and TECHNOLOGY, Vo., pp (011) ARTICLE Numerica Study on Subcooed Poo Boiing Yasuo OSE * and Tomoaki KUNUGI Kyoto Uniersity, Yoshida, Sakyo, Kyoto, , Japan This study focuses on the carification of the heat transfer characteristics of the subcooed poo boiing, the discussion on its mechanism, and the estabishment of a boiing and condenion mode for direct numerica simuation on the subcooed poo boiing phenomena. In this paper, three dimensiona numerica simuations based on the MARS (Muti-interface Adection and Reconstruction Soer) with a boiing and condenion mode which consisted of the improed phase-change mode and the reaxation time based on the quasi-therma equiibrium state hae been conducted for the bubbe growth process in the subcooed poo boiing. The numerica resuts regarding the bubbe growth process of the subcooed poo boiing show in good agreement with the experimenta obseration resuts and the existing anaytica equations among Rayeigh, Pesset and Zwick and Mikic et a. Therefore, it was found that the improed boiing and condenion mode with the reaxation time consideration can predict the bubbe growth process of the subcooed poo boiing phenomena. KEYWORDS: subcooed poo boiing, numerica simuation, phase-change, boiing and condenion mode, bubbe growth, quasi-therma equiibrium state I. Introduction 1 Boiing phenomena is a key to remoe the heat from the fue rods in nucear reactors such as BWR (Boiing Water Reactor) because the boiing heat transfer has most distinguished efficiency which can be enormous heat transfer coefficient compared to the conectie heat transfer of singe-phase fows. It wi aso pay a significant roe of the power generation efficiency in nucear reactors. Therefore, the mechanism of boiing phenomena has been studied extensiey oer the decades. This study focuses on the subcooed poo boiing in many boiing phenomena. Since the subcooed poo boiing is occurred under a condition beow the uration temperature, it is the most compicated phenomenon which incudes not ony the conectie heat transfer but aso the eaporation and condenion processes. Athough the subcooed boiing is ery important phenomena, the essentia mechanism has not yet been carified unti now because the bubbe nuceation and growth processes are too fast to obsere een by using the current high speed camera. Another approach to understand these processes is the numerica simuation. Since the prediction of boiing phenomena is ery important for the therma designs and managements, it has been proposed numerous prediction modes based on the experimenta data and/or theoretica considerations. Howeer, there is no "direct" numerica simuation mode for the subcooed boiing because of the ack of enough experimenta databases and theoretica considerations: here, "direct" means "without empirica correation." In recent years, with great adances in computer, numerica simuations for directy computing the bubbe dynamics regarding the nuceate boiing hae been performed by seera inestigators (Lee and Nydah; Wech; Son et a.; Yoon et a.; Shin et a.; Son and Dhir). 1-6) Howeer, they ony performed the numerica simuation of the urated poo boiing. Moreoer, it is not cear whether the numerica simuations with a high accuracy interface tracking for the arge bubbe deformation can be possibe. In this situation, Kunugi et a. carried out the three-dimensiona poo and forced conectie subcooed fow boiing phenomena by the MARS (Muti-interface Adection and Reconstruction Soer) which is based on a high-accuracy interface oume-tracking procedure. 7-8) In this study, it is focused on the carification of the heat transfer characteristics of the subcooed poo boiing, the discussion on its mechanism, and the estabishment of a boiing and condenion mode for the direct numerica simuation on the subcooed poo boiing phenomena. In this paper, the boiing and condenion mode is improed by introducing the foowing modes based on the quasi-therma equiibrium state: (1) an improed phase-change mode which consisted of the enthapy method for the water-apor system, () a reaxation time deried by considering the unsteady heat conduction. After that, three dimensiona numerica simuations based on the MARS with the improed boiing and condenion mode were performed for the bubbe growth process in the subcooed poo boiing, and then the resuts of the numerica simuations were compared with the experimenta obseration resuts and the existing anaytica equations among Rayeigh, Pesset and Zwick and Mikic et a. *Corresponding author, E-mai: ose@nuceng.kyoto-u.ac.jp c 011 Atomic Energy Society of Japan, A Rights Resered. 15

2 16 Yasuo OSE et a. II. Improement of Boiing and Condenion Mode Nuceation of a boiing bubbe needs to be modeed because the nuceation process is not competey understood today. The boiing and condenion mode in the MARS for the subcooed nuceate boiing phenomena consists of both a nuceation mode and a bubbe growth-condenion mode. 7) The nuceation bubbe in the boiing and condenion mode can be introduced by the critica nuceation bubbe based on the homogeneous nuceation theory in the superheated iquid at the meta-stabe state. In the nuceation mode, a homogeneous superheat imit of iquid, T SH gies the size of an embryo of the nuceation bubbe. T SH can be obtained by the kinetic theory. 9) A critica radius r e of the embryo corresponding to T SH can be cacuated by Eq. (1) based on thermodynamics. A computationa ce haing temperature oer T SH can be gien a VOF fraction of the embryo. Athough T SH may hae the spatia ariation on the heated surface according to the experiment, 10) T SH is assumed to be uniform on the heated surface, i.e., a nuceation site density is not considered in the present study. 11) re P ( T ) exp σ { [ P P ( T )]/ RT } ; ( T T P where σ is surface tension, T is temperature of iquid, P is pressure corresponding uration condition, P is pressure of iquid, is specific oume of iquid, and R is an idea gas constant per unit mass basis. The bubbe growth-condenion mode is based on the temperature-recoery method which is the improed enthapy method. 1) This mode is appied to ony the ce which has VOF fraction of both gas and iquid phases, i.e., the interfacia ce. Howeer, the origina mode coud not treat a arge oume change in the expansion and condenion processes because the temperature-recoery method has been deeoped for the soidification/meting of metas not for the water-apor phase-change system. Therefore, a density-change between water and apor was considered as a oume-change by a phase-change rate Δg, is expressed as: Δg SH ) (1) ρc pδt Sensibe heat ρ h Latent heat. () g Here, ρ is density, C p is specific heat at constant pressure, ΔT is degree of wa superheat, h is atent heat and the suffixes of g and denote gas and iquid phases, respectiey. Equation () means that the ratio of the sensibe heat to the atent heat at the interfacia ce. In order to isfy the conseration of the oume, Δg was incuded into F at condenion, or F g at eaporation. Here, F is oume of fuid (VOF) fraction. The origina bubbe growth-condenion mode is based on the assumptions of both a zero-thickness interface and a "rapid" change of "Sate 1: Water" to "State : Vapor" or ice ersa based on the quasi-therma equiibrium hypothesis. In contrast, a "ery sow" change of "State 1" to "State " in the quasi-therma equiibrium hypothesis is ignored. In the reaity, the finite thickness of interface exists, and both the "ery sow" and "rapid" changes may simutaneousy occur in the phase-change process. In order to consider a reaxation or waiting time for consuming the atent heat in the finite thickness interface region in the phase-change process, the unsteady heat conduction as the "ery sow" change process can be considered from the computationa-modeing point of iew as foows: The reaxation time t Δ can be introduced that the phase-change front passes through the computationa ce width Δ, so that t Δ can be defined by using the therma diffusiity of medium α as foows: t Δ /α. (3) Δ On the other hand, a therma penetration ength δ for a semi-infinite sab with a constant boundary temperature is approximated by the foowing expression: δ 1 αt Δ. (4) Substituting t Δ into Eq. (4), δ 1Δ. As the resut, an inariant reation between the therma penetration ength and the computationa ce width can be obtained as foows: δ Δ 1 δ 1 ( 1) (5) Therefore, the phase-changed oume during t Δ wi be 70% of the computationa ce, not 100%. This means the "ery sow" change can be reaized by this inariant constraint. In this paper, this inariant is defined as the reaxation time, and it can be considered if a VOF imiter is introduced as the phase-change judgment. For exampe, the VOF imiter (i.e., reaxation time) for both phase fronts is assumed to be ±15%, respectiey F 0.85 (6) III. Numerica Simuation Three dimensiona numerica simuations based on the MARS with the improed boiing and condenion mode based on the quasi-therma equiibrium hypothesis are performed and compared to the isuaization experiments in case of the degree of subcooing of 10.3 K for the bubbe growth process in the subcooed poo boiing. 13) Here, the isuaization experiment in the subcooed poo boiing was conducted by using the high-speed ideo camera (Phantom 7.1) mounted on a ong-focus microscope system. The fame rate of recording was 10,000 60,000 frames per second. The computationa domain for the bubbe growth process of the nuceation bubbe is shown in Fig. 1. In order to represent the nuceate boiing bubbe, smaer computationa grid must be needed: the grid size of 1μm in x-, y- and z-directions were used, respectiey. The computationa domain size was set to 60 μm (Length) 65 μm (Width) 60 μm (Height). The periodic boundary conditions were imposed at the x- and z-directions. The non-sip eocity condition was appied to the wa, and the upper boundary condition in y-direction was set to a constant pressure. In order to cacuate the soid heat conduction in the heating surface, a soid wa of 3 μm in thickness which simuated the patinum wire used in the experiment was ocated at the bottom of computationa domain. The constant heat fux of 0.5 MW/m from PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

3 Numerica Study on Subcooed Poo Boiing 17 Bubbe oume [mm 3 ] Experiment Rayeigh (Eq. 7) Origina mode Present mode Time [ms] Fig. Time ariation of bubbe oume in bubbe growth process at T sub 10.3 K Fig. 1 the outside was appied to the soid bottom wa. The initia system pressure was set to an atmospheric pressure and the degree of subcooing in the water poo was set to 10.3 K. The graitationa force was considered as the same as the experimenta condition. The hemisphere shaped embryo was put at the center of the heated surface as the initia condition. The superheated imit T SH was set to 383 K (110 degree C) which was estimated by using the waiting time of the bubbe generation cyce obtained from the experiment and the anaytica soution of the unsteady heat conduction, so that the critica diameter of the embryo by the nuceation mode was obtained about 6 μm. Time increment in the computation was set to 10 ns. IV. Resuts and Discussions Figure shows the time ariation of the bubbe oume change as a doube ogarithmic pot regarding the bubbe growth process. The square symbos depict the experimenta resuts at ΔT sub 10.3 K. The broken ine denotes the numerica resuts based on origina boiing and condenion mode and the soid ine denotes the numerica resuts obtained by the improed mode. Here, the imitation of the bubbe oume change existed because of the imitation of the computationa domain size. Since the bubbe growth is ery fast, the experimenta resuts in the beginning of the bubbe growth process can be considered as the inertia-controed one. It was aso known as the Rayeigh equation regarding a spherica bubbe growth in the homogeneous superheated iquid as foows: 14) r( t) Computationa domain for bubbe growth process T T 3 T hρ g ρ 1/ t. (7) Here, r is bubbe radius, T is temperature of the superheated ayer, T is uration temperature and t is time. The dotted ine in Fig. denotes the Rayeigh equation. It seems that the beginning of the bubbe growth process obtained by the experiment can be predicted by the Rayeigh equation. The present numerica resut (the soid ine) is aso in good agreement with the Rayeigh equation compared to the origina mode (the broken ine). This suggests the present boiing and condenion mode may hae a potentia to predict the bubbe growth process. Since the numerica simuations performed throughout the bubbe growth process was ery difficut because of the spatia- and tempora-scae changes during the bubbe growth process, it requires a tremendous computationa time and memory if the fix grid size (1 μm) is used for the whoe computation. Therefore, in order to further progress the numerica simuations for the bubbe growth process, the patch-work computations with changing the grid size are performed in this paper as foows: (1) The computation for the beginning of the bubbe growth process using the finest grid size of 1 μm at first. () Next, the bubbe oume obtained from the fina resut of the preious computation puts a hemisphere as the initia bubbe on the arger computationa domain with the grid size of 5 μm. Here, the initia temperature fied is recacuated without the bubbe. (3) To proceed the computation unti the top of the bubbe reaches to the ceiing of the computationa domain. (4) The same procedure repeats on much arger computationa domain with the grid size of 10 μm. On the other hand, according to the preious studies, the ater stage of bubbe growth process can be considered as the heat-transfer controed bubbe growth process. It was aso known that the existing anaytica equation proposed by Pesset and Zwick as foows: 15) ( T T ) 3 ρc p r( t) Ja α t, Ja. (8) π ρ h VOL., OCTOBER 011

4 18 Yasuo OSE et a. Bubbe oume [mm 3 ] In addition, the approximation equation containing both equations of 7 and 8 were proposed by Mikic et a. as foows: 16) + + ( 1) 3/ + r t + ( t ) 3 A (a) [ ] 3/ 1 [ T T ] h 3ρ T (b) (c) ρ 1/ Experiment Rayeigh (Eq. 7) Pesset and Zwick (Eq. 8) Mikic et a. (Eq. 9) Patchwork computations (Grid size1, 5, 10 μm) Time [ms] Fig. 3 Comparison of numerica resuts with experimenta resuts and existing anaytica equations in bubbe growth process at T sub 10.3 K + r( t), r, t B / A 1, B Ja α π 1/ + B t / A In this paper, the numerica resuts by the patch-work computation are compared to the experimenta resuts and the existing anaytica equations among Rayeigh (Eq. (8)), Pesset and Zwick (Eq. (8)) and Mikic et a. (Eq. (9)) in bubbe growth process. Figure 3 shows the time ariation of bubbe oume change as a singe ogarithmic pot regarding the bubbe growth process. The square symbo shows the experimenta resuts and the soid ine denotes the numerica resuts by the patch-work computations with changing the grid size of 1, 5 and 10 μm in a directions. The dotted ine denotes the Rayeigh equation (Eq. (7)), the singe-dotted dashed ine denotes the Pesset and Zwick equation (Eq. (8)) and the doube-dotted dashed denotes the equation of Mikic et a. (Eq. (9)). In these existing anaytica equations, since the infuence of the degree of subcooing on the bubbe growth process is appeared in the ater stage because the nuceate bubbe generates in the superheated ayer near the wa, the temperature of superheated ayer T can be estimated by using the experimenta resuts, i.e., 383 K by Eq. (7), 386 K by Eq. (8) and 393 K by Eq. (9). The summary of the comparison among them in Fig. 3 are as foows: (1) The numerica resuts with the grid size of 1μm for the beginning of the bubbe growth process as the inertia-controed process as shown in Fig. 3 (a) are in good agreement with the equations of Rayeigh and Mikic et a. () The numerica resuts with the grid size of 5μm as shown in Fig. 3 (b) are in good agreement with the experimenta resuts and the equation of Mikic et a., and show a itte bit apart from the Rayeigh, (9) equation. This means that the bubbe growth process is graduay changed from the inertia-controed process to the heat-transfer controed one. (3) The numerica resuts with the grid size of 10 μm as shown in Fig. 3 (c) are aso in good agreement with the experimenta resuts and the equation of Mikic et a., and are cose to the Pesset and Zwick equation as the heat-transfer controed bubbe growth process. Consequenty, it was found that the present boiing and condenion mode can retriee the experimenta resuts and the existing anaytica equations for both the beginning and the ater stages of the bubbe growth process. V. Concusions The numerica simuations based on the MARS with the improed boiing and condenion mode based on the quasi-therma equiibrium hypothesis were conducted for the bubbe growth process. The resuts of numerica simuations were compared with the experimenta resuts and the anaytica equations among Rayeigh, Pesset and Zwick, and Mikic et a. As the resuts, the numerica resuts of both the beginning and the ater stages of the bubbe growth process were in good agreement with the experimenta resuts and the existing anaytica equations. Therefore, it is concuded that the improed boiing and condenion mode with the reaxation time consideration can predict the bubbe growth process of the subcooed poo boiing phenomena. Acknowedgment This work was party supported by a Energy Science in the Age of Goba Warming of Goba Center of Exceence (G-COE) program (J-051) of the Ministry of Education, Cuture, Sports, Science and Technoogy of Japan. References 1) R. C. Lee, J. E. Nydah, Numerica cacuation of bubbe growth in nuceate boiing from inception through departure, J. Heat Trans., 111, (1989). ) S. W. J. Wech, Direct simuation of bubbe growth, J. Heat Trans., 41, (1998). 3) G. Son, V. K. Dhir, N. Ramanujapu, Dynamics and heat transfer associated with a singe bubbe during nuceate boiing on a horizonta surface, J. Heat Trans., 11, (1999). 4) H. Y. Yoon, S. Koshizuka, Y. Oka, Direct cacuation of bubbe growth, departure, and rise in nuceate poo boiing, Int. J. Mutiphase Fow, 7, (001). 5) S. Shin, S. I. Abde-Khaik, D. Juric, Direct three-dimensiona numerica simuation of nuceate boiing using the ee contour reconstruction method., Int. J. Mutiphase Fow, 31, (005). 6) G. Son, V. K. Dhir, Numerica simuation of nuceate boiing on a horizonta surface at high heat fuxes., Int. J. Heat Mass Tran., 51, (008). 7) T. Kunugi, N. Saito, T. Fujita, A. Serizawa, Direct numerica simuation of poo and forced conectie fow boiing phe- PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

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