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Flow instability analysis in U-tubes of steam generator under rolling movement
Annals of Nuclear Energy 64 (2014) 295–300
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Flow instability analysis in U-tubes of steam generator under rolling movement Jianli Hao a, Wenzhen Chen a,b,⇑, Xi Chu a, Shaoming Wang a a b
Department of Nuclear Energy Science and Engineering, Naval University of Engineering, Wuhan 430033, PR China Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, PR China
a r t i c l e
i n f o
Article history: Received 18 July 2013 Received in revised form 10 October 2013 Accepted 11 October 2013 Available online 8 November 2013 Keywords: Marine steam generator Natural circulation Flow instability Rolling movement
a b s t r a c t The reverse flow in U-tubes under natural circulation is a typical Ledinegg-type flow instability. Under rolling movement, the flow instability in U-tubes of marine steam generator is affected by an additional force, which makes the phenomenon more complicated. In the present work, a one-dimensional thermal hydraulic model for U-tubes under the natural circulation and rolling movement conditions is established. By the small perturbation theory, the critical pressure drop (CPD) and critical mass flow rate (CMFR) are proposed and analyzed, which determine the occurrence of reverse flow in U-tubes. The effects of rolling period and amplitude on the flow instability in U-tubes with different length are discussed. It is found that the fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but it is equal to half of the rolling period under the longitudinal swing movement. The relationships between the fluctuation amplitude of the CPD and rolling amplitude are in the form of linear and square functions under the transverse and longitudinal swing movements, respectively. The transverse swing movement has some effect on the spatial distribution of reverse flow, but the longitudinal swing movement has little effect on that parameter. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction During the natural circulation experiment (Kukita et al., 1988; Wang et al., 2007), it was found that the flow in U-tubes of steam generator (SG) is unstable and the reverse flow occurs within some U-tubes, which has negative effect on the operation of nuclear power plant. Sanders (1988) thought that the flow instability in U-tubes is a typical Ledinegg-type flow instability which can be studied by deriving the dynamic curve of pressure drop with velocity in the U-tubes. Yang et al. (2010) developed a lumped-distribution model to calculate the reverse flow in inverted U-tubes. Wang and Yu (2010) studied the space distribution of reverse flow as well as the effect of U-tube length on reverse flow by RELAP/MOD3.3. Zhang et al. (2011) found that the relationship between the U-tube length and characteristic pressure drop is nonlinear. But in these studies, the flow instability phenomenon in the U-tubes is mainly based on the land-based nuclear power plant, and the effects of ocean conditions on the flow instability phenomenon in the U-tubes of marine SG have not yet been studied.
⇑ Corresponding author. Present address: Department of Nuclear Energy Science and Engineering, Naval University of Engineering, Wuhan 430033, PR China. Tel.: +86 027 13871167436. E-mail address: [email protected] (W. Chen). 0306-4549/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2013.10.013
The effects of ocean conditions on the thermal hydraulic characteristic of nuclear power plant operated under natural circulation have attracted much concern due to its application background (Lewis, 1967; Chu et al., 2013). The researchers found that the flow and heat transfer in the marine nuclear power plant have different characteristic from that in the land-based nuclear power plant due to the heaving and rolling movement (Chen et al., 2012; Hao et al., 2012). In this paper, based on the small perturbation analysis, the critical pressure drop (CPD) and critical mass flow rate (CMFR) under rolling movement are obtained and analyzed, which determine the occurrence of reverse flow in U-tubes. And the effects of rolling period and rolling amplitude on the flow instability in U-tubes with different length are discussed.
2. Flow characteristic in U-tubes of SG under rolling movement Fig. 1 shows the schematic of the U-tube in SG under rolling movement. The flow and heat transfer in U-tubes can be simplified as follows (Sanders, 1988; Hao et al., 2013a): (1) one-dimensional approach is sufficient for modeling the flow in the U-tube; (2) the effects of rolling movement are considered only in the movement equation; (3) the density fluctuation caused by the temperature fluctuation can be ignored, which means the mass flow rate along the U-tube is not changed; (4) the water density in the U-tube can
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
Cold side
Hot side
Hot side
Cold side
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 1. Schematic of rolling movement.
be expressed as a linear function of temperature; and (5) the water in the secondary side of SG is in the saturated state. Based on Boussinesq approximation, the density can be written as follows (Hao et al., 2013b):
q ¼ q0 ½1 bðT T w Þ
_ @ q @ðmÞ þ ¼0 A@s @t
ð2Þ
_ _2 l @m fl m ¼ Dp þ DqgH 2 þ K þ Dpa A @t d0 2A q
ð3Þ
1 @q @q Phsp þ ¼ ðq q Þ _ p 0 u @t @s mC
ð4Þ
where t is the time, s is the coordinate in the normal flow direction _ along the U-tube, A is the U-tube flow area, l is the U-tube length, m is the mass flow rate, Dp is the pressure drop in U-tubes, Dq is the density difference between the cold side and hot side of the U-tube, is the g is the gravitational acceleration, H is the U-tube height, q average density, f is the friction resistance coefficient, f = 0.3164Re0.25, Re is the Reynolds number, K is the local resistance coefficient, K = 0.262 + 0.326(d0/ru)3.5, d0 is the U-tube inside diameter, ru is the U-tube bending radius, Dpa is the additional pressure drop caused by the rolling movement, u is the velocity, P is the wetted perimeter of U-tube, hsp is the overall heat transfer coeffi1 , h0 is the surface heat transfer coefficient, hsp ¼ ln
d1 d0
Dpa ¼
l
b¼
T2
0
! where F a is the additional force caused by the rolling movement, which can be expressed by:
~ ðx ~ ~ ~ ~ b ~ rÞ þ ~ r þ 2x u F a ¼ q½x
ð6Þ
~ is the angular velocity, ~ b is the angular acceleration, ~ r is where x the vector radius. It is usually assumed that the rolling movement
2pt T
ð8Þ
q ¼ q0 Dq0 eA1 s
ð9Þ
_ p , Dq0 =.q0 qin. where A1 ¼ hsp P=mc . R Rl R ¼ 0l q ds l and Dq ¼ l=2 From Eq. (9), q q ds 0l=2 q ds l=2 can be derived as follows:
q ¼ q0 Dq ¼
Dq0 1 eA1 l lA1
ð10Þ
A1 l 2 2Dq0 1 e 2 A1 l
ð11Þ
Combining Eqs. (5), (6), and (9), the additional pressure drop caused by the rolling movement can be derived as follows:
Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ Dpa;s ¼ bRl q0 ð1 eA1 l Þ A1 lA1 þ
Dpa;l ¼
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
ð12Þ
A21
x2 r2u Dq0 eA1 H0 4þ 2
þ ð5Þ
cos
ð7Þ
where hM and T are the rolling amplitude and period, respectively. _ and fluid density (qin) are giWhile the inlet mass flow rate ðmÞ ven, based on Eqs. (2) and (4), the fluid density can be derived as follows:
d0 1 d1
! s F a d~
2phM 2pt sin T T
4p2 hM
þh
cient of U-tube inside wall, kwall is the U-tube thermal conductivity, d1 is the U-tube outside diameter, h1 is the surface heat transfer coefficient of U-tube outside wall. Dpa can be given as follows (Lewis, 1967; Chu et al., 2013):
Z
x¼
ð1Þ
where q is the density, b is the thermal expansion coefficient, q0 and Tw are the reference density and temperature, respectively. The conservation equations in U-tubes can be written as follows (Chen et al., 2013):
1 þ d0 h0 2kwall
is in a sinusoidal order. The angle velocity and acceleration can be written as follows (Chen et al., 2013):
r 2u A21
ð1 eA1 pru Þ
x Dq0 ð1 e
A1 H0
A1 ðH0 þpr u Þ
Þ½1 þ e
A21
x2 Dq0 H0 ð1 þ eA1 l Þ A1
ð13Þ
where subscript s and l represent the transverse and longitudinal swing movement, respectively, H0 is the height of straight tube, H0 = H ru. _ is given, substituting Eqs. While the inlet mass flow rate ðmÞ (10)–(13) into Eq. (3), the variations of pressure drop with mass flow rate in the U-tube under the rolling movement can be derived as follows:
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Dps ¼
_2 m
A1 l 2 2Dq0 ð1 e 2 Þ gH A1 l ð1 eA1 l Þ 2A q0 Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ ð1 eA1 l Þ þ bRl q0 A1 lA1
ðfl=d0 þ KÞ Dq0 lA1
2
The flow is stable while ki > 0, and the flow is unstable while ki < 0. ki = 0 corresponds to the flow instability criteria, which can be obtained from Eqs. (18) and (19) as follows:
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
_ c;s ¼ m
q0 DlAq10 ð1 eA1 l Þ Dq0 cp g ðfl=d0 þ KÞ=A
ð14Þ
A21
4þ
r2u A21
ð1 eA1 pru Þ þ
x2 Dq0 H0 ð1 þ eA1 l Þ A1
_ c;l ¼ m
x2 Dq0 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ
ð15Þ
A21
To study the flow instability, a small disturbed mass flow rate is considered as follows:
_ _ 0 þ dðtÞ mðtÞ ¼m
þ
2
ðfl=d0 þ KÞ=A
dd ¼ ki d i ¼ s; l dt
ð17Þ
where ks and kl are corresponding to the transvers
( A1 l 2 _0 A m ðfl=d0 þ KÞ Dq c p g 0 ð1 e 2 Þ D q 2 l A q0 0 ð1 eA1 l Þ hsp P lA1
x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0
)
2
ðhsp P=cp Þ =2 ð19Þ
Integrating Eq. (17) results in:
d ¼ d0 eki t
ð20Þ
hsp P
2
x2 H0 Dq0 ð1 þ eA1 l Þ hsp P=cp
)
ð22Þ
2
A1 l 2 2Dq0 1 e 2 gH A1 l ð1 2A q0 Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ ð1 eA1 l Þ þ bRl q0 A1 lA1 _ 2c m
Dpc;l ¼
ðfl=d0 þ KÞ Dq0 lA1
2
eA1 l Þ
ð23Þ
A21 ðfl=d0 þ KÞ
Dq 2A2 q0 lA10 ð1 eA1 l Þ
x2 r2u Dq0 eA1 H0 4þ 2
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
_ 2c m
(
ð18Þ
A1 l
1 e 2
ðhsp P=cp Þ =2
Dq br A1 l 2 _0 A m ðfl=d0 þ KÞ Dq cp g u 0 ks ¼ 1 e 2 þ 0 ð1 eA1 l Þ 2 D q l A q0 0 ð1 eA1 l Þ hsp P hsp P=cp lA1 ) x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0 þ 2 hsp P=cp ðhsp P=cp Þ =2
2
_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0 m
Dpc;s ¼
ð16Þ
)
_ c is the critical mass flow rate (CMFR). where m Substituting Eqs. (21) and (22) into Eqs. (14) and (15), respectively, the critical pressure drop (CPD) under the transverse and longitudinal swing movement in the U-tube can be obtained as follows:
_ 0 is the steady mass flow rate, d(t) is the small disturbance. where m Substituting Eq. (16) into Eq. (3) and eliminating the steady terms, the following equation can be derived:
hsp P=cp
Dq0 bru ð1 eA1 l Þ hsp P=cp
ðhsp P=cp Þ =2
q0 DlAq10 ð1 eA1 l Þ Dq0 cp g
3. Flow instability analysis in U-tubes
þ
ð21Þ
where Dps and Dpl are the pressure drop under the transverse and longitudinal swing movement, respectively. While @@Dm_p < 0, the flow in U-tubes is unstable, and the reverse flow will occur. In Section 3, this flow instability in U-tubes will be explained by the small perturbation theory.
kl ¼
þ
hsp P=cp
ðfl=d0 þ KÞ
x2 r2u Dq0 eA1 H0
hsp P
2
x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0
A1 l 2 2Dq0 Dpl ¼ 2 1 e 2 gH Dq0 A1 l 2A q0 lA1 ð1 eA1 l Þ _2 m
2
A1 l
1 e 2
r 2u A21
x Dq0 ð1 e
A1 l 2 2Dq0 1 e 2 gH A1 l
ð1 eA1 pru Þ þ
A1 H0
x2 Dq0 H0 ð1 þ eA1 l Þ
A1 ðH0 þpr u Þ
Þ½1 þ e
A1
ð24Þ
A21
While the pressure drop in the U-tube is lower than Dpc, the flow in the U-tube is unstable, and the reverse flow will occur. The critical pressure drop (CPD) is the flow instability criterion, which is discussed in the following section. 4. Numerical experiments In order to validate the effect of rolling movement on the flow instability in U-tubes, a certain type of marine steam generator is selected (Hao et al., 2013a, 2014), the parameters of U-tubes are shown in Table 1, where l0 is the longest tube length of the marine steam generator. The inside and outside diameters and straight tube length of the U-tubes are the same, but the bending radiuses of the U-tubes are different.
Table 1 Parameters of marine steam generator U-tubes. Name of parameter
Value of parameter
Name of parameter
Value of parameter
Inside diameter of U-tube, m Length of U-tube, m Secondary side operating pressure of steam generator, MPa Inlet fluid temperature, K
d0 0.7l0 l0 3.0 Tin
Outside diameter of U-tube, m Primary side operating pressure of steam generator, MPa Mass flow rate, kg/s Gravitational acceleration, m/s2
d1 14.0 0.006–0.06 9.8
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
4.1. Variation curve of pressure drop with mass flow rate within U-tubes Here the U-tube length is chosen as 0.7 l0, the rolling movement period is 10 s, and the amplitude is 10°. The variations of pressure drop with mass flow rate for different rolling movement time are shown in Fig. 2. It can be seen from Fig. 2 that the rolling movement has a certain effect on the relationship between the pressure drop and mass flow rate. The mass flow rate and pressure drop at the inflection point of the curves in Fig. 2 are namely CMFR and CPD, respectively. The data in Fig. 2 are analyzed, and found that the rolling movement has large effect on CPD, but little effect on CMFR. The effects of rolling movement on CPD are discussed in the following section. 4.2. Effect of rolling period on CPD Here the U-tube length is chosen as 0.7l0, the rolling movement amplitude is 10°, and the period is 10, 15 and 20 s, respectively. Based on Eqs. (23) and (24), the effect of rolling period on CPD is analyzed, as shown in Fig. 3. From Fig. 3 it can be seen that the CPD changes periodically under the rolling movement, namely the CPD increases during the first half
(a) Transverse swing movement
period, then decreases during the last half period under the transverse swing movement, but the variation of CPD under the longitudinal swing movement is opposite to that. The fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but the fluctuation period of CPD is the half of the rolling period under the longitudinal swing movement. As the rolling period increases, the fluctuation amplitude of CPD decreases. The transverse swing movement has larger effect on the fluctuation amplitude of CPD than the longitudinal swing movement. 4.3. Effect of rolling amplitude on CPD Here the U-tube length is chosen as 0.7l0, the rolling movement period is 10s, and the amplitude is 5°, 10°, 15° and 20°, respectively. Based on Eqs. (23) and (24), the effect of rolling amplitude on CPD is analyzed, as shown in Fig. 4. It can be seen from Fig. 4 that the CPD fluctuation amplitude (DpcM = (CPDmax CPDmin)/2) under the transverse swing movement is much larger than that under the longitudinal swing movement, and it increases with the increase of rolling amplitude. The data in Fig. 4a and b are fitted, respectively, and then the variation of the CPD fluctuation amplitude (DpcM) with rolling amplitude can be expressed as follows:
(b) Longitudinal swing movement
Fig. 2. Variation of pressure drop with mass flow rate for different time.
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 3. Variation of critical pressure drop with time for different rolling period.
J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
(a) Transverse swing movement
299
(b) Longitudinal swing movement
Fig. 4. Variation of critical pressure drop with time for different rolling amplitude.
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 5. Variation of critical pressure drop with time in U-tubes with different length under rolling movement.
DpcM;s ¼ 3:1588hM
ð25Þ
DpcM;l ¼ 0:0076h2M
ð26Þ
From Eqs. (25) and (26), we can see that the relationships between the CPD fluctuation amplitude and rolling amplitude are the linear and square functions under the transverse and longitudinal swing movement, respectively. 4.4. Effect of rolling movement on flow instability in U-tubes with different length Based on Eqs. (23) and (24), the flow instability in U-tubes with different length under the rolling movement are investigated, and the results are shown in Fig. 5. In Fig. 5, the rolling movement period is 10s, and the amplitude is 10°, l1 = 0.70l0, l2 = 0.85l0 and l3 = 1.00l0. l1 and l3 are the shortest and longest length of U-tubes in the marine steam generator, respectively. It can be seen from Fig. 5a that under the transverse swing movement the CPD of U-tubes with the shortest length is higher than the others at the first quarter time, and the CPD of U-tubes with the longest length is higher than the others at the second
and third quarter time. As the pressure drop of the parallel U-tubes is the same, at the first quarter time, the shortest U-tubes reach their CPD earlier, which means the reverse flow will occur in the short U-tubes easily. On the contrary, at the second and third quarter time, the reverse flow will occur in the long U-tubes easily. The transverse swing movement has an obvious effect on the space distribution of reverse flow. From Fig. 5b, it can be seen that the variations of CPD with time in U-tubes with different length are similar, and the CPD of U-tubes with shortest length is higher than the others, which means the reverse flow will occur in the short U-tube easily. The longitudinal swing movement has little effect on the space distribution of reverse flow.
5. Conclusions Under rolling movement and natural circulation conditions, one-dimensional thermal hydraulic model in U-tubes is established. By the small perturbation theory, the critical pressure drop (CPD) and critical mass flow rate (CMFR) are given and analyzed, which determine the occurrence of reverse flow in U-tubes. The effect of rolling movement on the flow instability in U-tubes is discussed, and the detailed conclusions are drawn as follows:
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
(1) The rolling movement has a certain effect on the relationship between pressure drop and mass flow rate. (2) The CPD changes periodically under the rolling movement, and the fluctuation amplitude of CPD decreases with the increase of rolling period. The fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but it is the half of the rolling period under the longitudinal swing movement. The transverse swing movement has larger effect on the fluctuation amplitude of CPD than the longitudinal swing movement. (3) The relationship between the fluctuation amplitude of CPD and rolling amplitude is linear function under the transverse swing movement, and it is square function under the longitudinal swing movement. (4) The transverse swing movement has an obvious effect on the space distribution of reverse flow, but the longitudinal swing movement has little effect on the space distribution of reverse flow.
Acknowledgement This research is supported by the Doctorial Innovation Fund of Naval University of Engineering. References Chen, Z.Y., Hao, J.L., Chen, W.Z., 2012. The development of fast simulation program for marine reactor parameters. Ann. Nucl. Energy 40, 45–52.
Chen, W.Z., Yu, L., Hao, J.L., 2013. Thermal Hydraulics of Nuclear Power Plants. Chinese Atomic Energy Press, Beijing (in Chinese). Chu, X., Chen, W.Z., Hao, J.L., 2013. Analysis of reverse flow in U-tubes of UTSG in rocking motion. Prog. Nucl. Energy. http://dx.doi.org/10.1016/ j.pnucene.2013.10.017. Hao, J.L., Chen, W.Z., Chen, Z.Y., 2012. The development of natural circulation operation support program for ship nuclear power machinery. Ann. Nucl. Energy 50, 199–205. Hao, J.L., Chen, W.Z., Zhang, D., 2013a. Effect of U-tube length on reverse flow in UTSG primary side under natural circulation. Ann. Nucl. Energy 56, 66–70. Hao, J.L., Chen, W.Z., Zhang, D., Wang, S.M., 2013b. Scaling modeling analysis of flow instability in U-tubes of steam generator under natural circulation. Ann. Nucl. Energy 64, 169–175. Hao, J.L., Chen, W.Z., Wang, S.M., 2014. Flow instability analysis of U-tubes in SG based on CFD method. Prog. Nucl. Energy 70, 134–139, http://dx.doi.org/ 10.1016/j.pnucene.2013.08.010. Kukita, Y., Nakamura, H., Tasaka, K., 1988. Nonuniform steam generator U-Tube flow distribution during natural circulation tests in ROSA-IV large scale test facility. Nucl. Sci. Eng. 99, 289–298. Lewis, E.V., 1967. The Motion of Ships in Waves. Principles of Naval Architecture, New York. Sanders, J., 1988. Stability of single-phase natural circulation with inverted U-tube steam generators. J. Heat Transfer 110, 735–742. Wang, C., Yu, L., 2010. Asymmetry investigation on single phase flow in inverted Utubes of steam generator under the condition of natural circulation. In: ICONE18-29502, Xi’an, China. Wang, F., Zhuo, W.B., Xiao, Z.J., et al., 2007. Phenomena and analysis of reversal flow in vertically inverted U-tube steam generator. Atom. Energy Sci. Technol. 41 (1), 65–68 (in Chinese). Yang, R.C., Liu, J.G., Huang, Y.P., et al., 2010. Calculation of reverse flow in inverted U-tubes of steam generator during natural circulation. Nucl. Power Eng. 31 (1), 57–60 (in Chinese). Zhang, D., Chen, W.Z., Wang, S.M., et al., 2011. Influence of U-tube length on space distribution of UTSG reverse flow in tube. Atom. Energy Sci. Technol. 45 (6), 667–671 (in Chinese).
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Flow instability analysis in U-tubes of steam generator under rolling movement Jianli Hao a, Wenzhen Chen a,b,⇑, Xi Chu a, Shaoming Wang a a b
Department of Nuclear Energy Science and Engineering, Naval University of Engineering, Wuhan 430033, PR China Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, PR China
a r t i c l e
i n f o
Article history: Received 18 July 2013 Received in revised form 10 October 2013 Accepted 11 October 2013 Available online 8 November 2013 Keywords: Marine steam generator Natural circulation Flow instability Rolling movement
a b s t r a c t The reverse flow in U-tubes under natural circulation is a typical Ledinegg-type flow instability. Under rolling movement, the flow instability in U-tubes of marine steam generator is affected by an additional force, which makes the phenomenon more complicated. In the present work, a one-dimensional thermal hydraulic model for U-tubes under the natural circulation and rolling movement conditions is established. By the small perturbation theory, the critical pressure drop (CPD) and critical mass flow rate (CMFR) are proposed and analyzed, which determine the occurrence of reverse flow in U-tubes. The effects of rolling period and amplitude on the flow instability in U-tubes with different length are discussed. It is found that the fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but it is equal to half of the rolling period under the longitudinal swing movement. The relationships between the fluctuation amplitude of the CPD and rolling amplitude are in the form of linear and square functions under the transverse and longitudinal swing movements, respectively. The transverse swing movement has some effect on the spatial distribution of reverse flow, but the longitudinal swing movement has little effect on that parameter. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction During the natural circulation experiment (Kukita et al., 1988; Wang et al., 2007), it was found that the flow in U-tubes of steam generator (SG) is unstable and the reverse flow occurs within some U-tubes, which has negative effect on the operation of nuclear power plant. Sanders (1988) thought that the flow instability in U-tubes is a typical Ledinegg-type flow instability which can be studied by deriving the dynamic curve of pressure drop with velocity in the U-tubes. Yang et al. (2010) developed a lumped-distribution model to calculate the reverse flow in inverted U-tubes. Wang and Yu (2010) studied the space distribution of reverse flow as well as the effect of U-tube length on reverse flow by RELAP/MOD3.3. Zhang et al. (2011) found that the relationship between the U-tube length and characteristic pressure drop is nonlinear. But in these studies, the flow instability phenomenon in the U-tubes is mainly based on the land-based nuclear power plant, and the effects of ocean conditions on the flow instability phenomenon in the U-tubes of marine SG have not yet been studied.
⇑ Corresponding author. Present address: Department of Nuclear Energy Science and Engineering, Naval University of Engineering, Wuhan 430033, PR China. Tel.: +86 027 13871167436. E-mail address: [email protected] (W. Chen). 0306-4549/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2013.10.013
The effects of ocean conditions on the thermal hydraulic characteristic of nuclear power plant operated under natural circulation have attracted much concern due to its application background (Lewis, 1967; Chu et al., 2013). The researchers found that the flow and heat transfer in the marine nuclear power plant have different characteristic from that in the land-based nuclear power plant due to the heaving and rolling movement (Chen et al., 2012; Hao et al., 2012). In this paper, based on the small perturbation analysis, the critical pressure drop (CPD) and critical mass flow rate (CMFR) under rolling movement are obtained and analyzed, which determine the occurrence of reverse flow in U-tubes. And the effects of rolling period and rolling amplitude on the flow instability in U-tubes with different length are discussed.
2. Flow characteristic in U-tubes of SG under rolling movement Fig. 1 shows the schematic of the U-tube in SG under rolling movement. The flow and heat transfer in U-tubes can be simplified as follows (Sanders, 1988; Hao et al., 2013a): (1) one-dimensional approach is sufficient for modeling the flow in the U-tube; (2) the effects of rolling movement are considered only in the movement equation; (3) the density fluctuation caused by the temperature fluctuation can be ignored, which means the mass flow rate along the U-tube is not changed; (4) the water density in the U-tube can
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
Cold side
Hot side
Hot side
Cold side
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 1. Schematic of rolling movement.
be expressed as a linear function of temperature; and (5) the water in the secondary side of SG is in the saturated state. Based on Boussinesq approximation, the density can be written as follows (Hao et al., 2013b):
q ¼ q0 ½1 bðT T w Þ
_ @ q @ðmÞ þ ¼0 A@s @t
ð2Þ
_ _2 l @m fl m ¼ Dp þ DqgH 2 þ K þ Dpa A @t d0 2A q
ð3Þ
1 @q @q Phsp þ ¼ ðq q Þ _ p 0 u @t @s mC
ð4Þ
where t is the time, s is the coordinate in the normal flow direction _ along the U-tube, A is the U-tube flow area, l is the U-tube length, m is the mass flow rate, Dp is the pressure drop in U-tubes, Dq is the density difference between the cold side and hot side of the U-tube, is the g is the gravitational acceleration, H is the U-tube height, q average density, f is the friction resistance coefficient, f = 0.3164Re0.25, Re is the Reynolds number, K is the local resistance coefficient, K = 0.262 + 0.326(d0/ru)3.5, d0 is the U-tube inside diameter, ru is the U-tube bending radius, Dpa is the additional pressure drop caused by the rolling movement, u is the velocity, P is the wetted perimeter of U-tube, hsp is the overall heat transfer coeffi1 , h0 is the surface heat transfer coefficient, hsp ¼ ln
d1 d0
Dpa ¼
l
b¼
T2
0
! where F a is the additional force caused by the rolling movement, which can be expressed by:
~ ðx ~ ~ ~ ~ b ~ rÞ þ ~ r þ 2x u F a ¼ q½x
ð6Þ
~ is the angular velocity, ~ b is the angular acceleration, ~ r is where x the vector radius. It is usually assumed that the rolling movement
2pt T
ð8Þ
q ¼ q0 Dq0 eA1 s
ð9Þ
_ p , Dq0 =.q0 qin. where A1 ¼ hsp P=mc . R Rl R ¼ 0l q ds l and Dq ¼ l=2 From Eq. (9), q q ds 0l=2 q ds l=2 can be derived as follows:
q ¼ q0 Dq ¼
Dq0 1 eA1 l lA1
ð10Þ
A1 l 2 2Dq0 1 e 2 A1 l
ð11Þ
Combining Eqs. (5), (6), and (9), the additional pressure drop caused by the rolling movement can be derived as follows:
Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ Dpa;s ¼ bRl q0 ð1 eA1 l Þ A1 lA1 þ
Dpa;l ¼
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
ð12Þ
A21
x2 r2u Dq0 eA1 H0 4þ 2
þ ð5Þ
cos
ð7Þ
where hM and T are the rolling amplitude and period, respectively. _ and fluid density (qin) are giWhile the inlet mass flow rate ðmÞ ven, based on Eqs. (2) and (4), the fluid density can be derived as follows:
d0 1 d1
! s F a d~
2phM 2pt sin T T
4p2 hM
þh
cient of U-tube inside wall, kwall is the U-tube thermal conductivity, d1 is the U-tube outside diameter, h1 is the surface heat transfer coefficient of U-tube outside wall. Dpa can be given as follows (Lewis, 1967; Chu et al., 2013):
Z
x¼
ð1Þ
where q is the density, b is the thermal expansion coefficient, q0 and Tw are the reference density and temperature, respectively. The conservation equations in U-tubes can be written as follows (Chen et al., 2013):
1 þ d0 h0 2kwall
is in a sinusoidal order. The angle velocity and acceleration can be written as follows (Chen et al., 2013):
r 2u A21
ð1 eA1 pru Þ
x Dq0 ð1 e
A1 H0
A1 ðH0 þpr u Þ
Þ½1 þ e
A21
x2 Dq0 H0 ð1 þ eA1 l Þ A1
ð13Þ
where subscript s and l represent the transverse and longitudinal swing movement, respectively, H0 is the height of straight tube, H0 = H ru. _ is given, substituting Eqs. While the inlet mass flow rate ðmÞ (10)–(13) into Eq. (3), the variations of pressure drop with mass flow rate in the U-tube under the rolling movement can be derived as follows:
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Dps ¼
_2 m
A1 l 2 2Dq0 ð1 e 2 Þ gH A1 l ð1 eA1 l Þ 2A q0 Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ ð1 eA1 l Þ þ bRl q0 A1 lA1
ðfl=d0 þ KÞ Dq0 lA1
2
The flow is stable while ki > 0, and the flow is unstable while ki < 0. ki = 0 corresponds to the flow instability criteria, which can be obtained from Eqs. (18) and (19) as follows:
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
_ c;s ¼ m
q0 DlAq10 ð1 eA1 l Þ Dq0 cp g ðfl=d0 þ KÞ=A
ð14Þ
A21
4þ
r2u A21
ð1 eA1 pru Þ þ
x2 Dq0 H0 ð1 þ eA1 l Þ A1
_ c;l ¼ m
x2 Dq0 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ
ð15Þ
A21
To study the flow instability, a small disturbed mass flow rate is considered as follows:
_ _ 0 þ dðtÞ mðtÞ ¼m
þ
2
ðfl=d0 þ KÞ=A
dd ¼ ki d i ¼ s; l dt
ð17Þ
where ks and kl are corresponding to the transvers
( A1 l 2 _0 A m ðfl=d0 þ KÞ Dq c p g 0 ð1 e 2 Þ D q 2 l A q0 0 ð1 eA1 l Þ hsp P lA1
x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0
)
2
ðhsp P=cp Þ =2 ð19Þ
Integrating Eq. (17) results in:
d ¼ d0 eki t
ð20Þ
hsp P
2
x2 H0 Dq0 ð1 þ eA1 l Þ hsp P=cp
)
ð22Þ
2
A1 l 2 2Dq0 1 e 2 gH A1 l ð1 2A q0 Dq0 x2 H0 Dq0 ð1 þ eA1 l Þ ð1 eA1 l Þ þ bRl q0 A1 lA1 _ 2c m
Dpc;l ¼
ðfl=d0 þ KÞ Dq0 lA1
2
eA1 l Þ
ð23Þ
A21 ðfl=d0 þ KÞ
Dq 2A2 q0 lA10 ð1 eA1 l Þ
x2 r2u Dq0 eA1 H0 4þ 2
x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpRÞ Dq0
_ 2c m
(
ð18Þ
A1 l
1 e 2
ðhsp P=cp Þ =2
Dq br A1 l 2 _0 A m ðfl=d0 þ KÞ Dq cp g u 0 ks ¼ 1 e 2 þ 0 ð1 eA1 l Þ 2 D q l A q0 0 ð1 eA1 l Þ hsp P hsp P=cp lA1 ) x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0 þ 2 hsp P=cp ðhsp P=cp Þ =2
2
_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0 m
Dpc;s ¼
ð16Þ
)
_ c is the critical mass flow rate (CMFR). where m Substituting Eqs. (21) and (22) into Eqs. (14) and (15), respectively, the critical pressure drop (CPD) under the transverse and longitudinal swing movement in the U-tube can be obtained as follows:
_ 0 is the steady mass flow rate, d(t) is the small disturbance. where m Substituting Eq. (16) into Eq. (3) and eliminating the steady terms, the following equation can be derived:
hsp P=cp
Dq0 bru ð1 eA1 l Þ hsp P=cp
ðhsp P=cp Þ =2
q0 DlAq10 ð1 eA1 l Þ Dq0 cp g
3. Flow instability analysis in U-tubes
þ
ð21Þ
where Dps and Dpl are the pressure drop under the transverse and longitudinal swing movement, respectively. While @@Dm_p < 0, the flow in U-tubes is unstable, and the reverse flow will occur. In Section 3, this flow instability in U-tubes will be explained by the small perturbation theory.
kl ¼
þ
hsp P=cp
ðfl=d0 þ KÞ
x2 r2u Dq0 eA1 H0
hsp P
2
x2 H0 Dq0 ð1 þ eA1 l Þ m_ 0 x2 ð1 eA1 H0 Þ½1 þ eA1 ðH0 þpru Þ Dq0
A1 l 2 2Dq0 Dpl ¼ 2 1 e 2 gH Dq0 A1 l 2A q0 lA1 ð1 eA1 l Þ _2 m
2
A1 l
1 e 2
r 2u A21
x Dq0 ð1 e
A1 l 2 2Dq0 1 e 2 gH A1 l
ð1 eA1 pru Þ þ
A1 H0
x2 Dq0 H0 ð1 þ eA1 l Þ
A1 ðH0 þpr u Þ
Þ½1 þ e
A1
ð24Þ
A21
While the pressure drop in the U-tube is lower than Dpc, the flow in the U-tube is unstable, and the reverse flow will occur. The critical pressure drop (CPD) is the flow instability criterion, which is discussed in the following section. 4. Numerical experiments In order to validate the effect of rolling movement on the flow instability in U-tubes, a certain type of marine steam generator is selected (Hao et al., 2013a, 2014), the parameters of U-tubes are shown in Table 1, where l0 is the longest tube length of the marine steam generator. The inside and outside diameters and straight tube length of the U-tubes are the same, but the bending radiuses of the U-tubes are different.
Table 1 Parameters of marine steam generator U-tubes. Name of parameter
Value of parameter
Name of parameter
Value of parameter
Inside diameter of U-tube, m Length of U-tube, m Secondary side operating pressure of steam generator, MPa Inlet fluid temperature, K
d0 0.7l0 l0 3.0 Tin
Outside diameter of U-tube, m Primary side operating pressure of steam generator, MPa Mass flow rate, kg/s Gravitational acceleration, m/s2
d1 14.0 0.006–0.06 9.8
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
4.1. Variation curve of pressure drop with mass flow rate within U-tubes Here the U-tube length is chosen as 0.7 l0, the rolling movement period is 10 s, and the amplitude is 10°. The variations of pressure drop with mass flow rate for different rolling movement time are shown in Fig. 2. It can be seen from Fig. 2 that the rolling movement has a certain effect on the relationship between the pressure drop and mass flow rate. The mass flow rate and pressure drop at the inflection point of the curves in Fig. 2 are namely CMFR and CPD, respectively. The data in Fig. 2 are analyzed, and found that the rolling movement has large effect on CPD, but little effect on CMFR. The effects of rolling movement on CPD are discussed in the following section. 4.2. Effect of rolling period on CPD Here the U-tube length is chosen as 0.7l0, the rolling movement amplitude is 10°, and the period is 10, 15 and 20 s, respectively. Based on Eqs. (23) and (24), the effect of rolling period on CPD is analyzed, as shown in Fig. 3. From Fig. 3 it can be seen that the CPD changes periodically under the rolling movement, namely the CPD increases during the first half
(a) Transverse swing movement
period, then decreases during the last half period under the transverse swing movement, but the variation of CPD under the longitudinal swing movement is opposite to that. The fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but the fluctuation period of CPD is the half of the rolling period under the longitudinal swing movement. As the rolling period increases, the fluctuation amplitude of CPD decreases. The transverse swing movement has larger effect on the fluctuation amplitude of CPD than the longitudinal swing movement. 4.3. Effect of rolling amplitude on CPD Here the U-tube length is chosen as 0.7l0, the rolling movement period is 10s, and the amplitude is 5°, 10°, 15° and 20°, respectively. Based on Eqs. (23) and (24), the effect of rolling amplitude on CPD is analyzed, as shown in Fig. 4. It can be seen from Fig. 4 that the CPD fluctuation amplitude (DpcM = (CPDmax CPDmin)/2) under the transverse swing movement is much larger than that under the longitudinal swing movement, and it increases with the increase of rolling amplitude. The data in Fig. 4a and b are fitted, respectively, and then the variation of the CPD fluctuation amplitude (DpcM) with rolling amplitude can be expressed as follows:
(b) Longitudinal swing movement
Fig. 2. Variation of pressure drop with mass flow rate for different time.
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 3. Variation of critical pressure drop with time for different rolling period.
J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
(a) Transverse swing movement
299
(b) Longitudinal swing movement
Fig. 4. Variation of critical pressure drop with time for different rolling amplitude.
(a) Transverse swing movement
(b) Longitudinal swing movement
Fig. 5. Variation of critical pressure drop with time in U-tubes with different length under rolling movement.
DpcM;s ¼ 3:1588hM
ð25Þ
DpcM;l ¼ 0:0076h2M
ð26Þ
From Eqs. (25) and (26), we can see that the relationships between the CPD fluctuation amplitude and rolling amplitude are the linear and square functions under the transverse and longitudinal swing movement, respectively. 4.4. Effect of rolling movement on flow instability in U-tubes with different length Based on Eqs. (23) and (24), the flow instability in U-tubes with different length under the rolling movement are investigated, and the results are shown in Fig. 5. In Fig. 5, the rolling movement period is 10s, and the amplitude is 10°, l1 = 0.70l0, l2 = 0.85l0 and l3 = 1.00l0. l1 and l3 are the shortest and longest length of U-tubes in the marine steam generator, respectively. It can be seen from Fig. 5a that under the transverse swing movement the CPD of U-tubes with the shortest length is higher than the others at the first quarter time, and the CPD of U-tubes with the longest length is higher than the others at the second
and third quarter time. As the pressure drop of the parallel U-tubes is the same, at the first quarter time, the shortest U-tubes reach their CPD earlier, which means the reverse flow will occur in the short U-tubes easily. On the contrary, at the second and third quarter time, the reverse flow will occur in the long U-tubes easily. The transverse swing movement has an obvious effect on the space distribution of reverse flow. From Fig. 5b, it can be seen that the variations of CPD with time in U-tubes with different length are similar, and the CPD of U-tubes with shortest length is higher than the others, which means the reverse flow will occur in the short U-tube easily. The longitudinal swing movement has little effect on the space distribution of reverse flow.
5. Conclusions Under rolling movement and natural circulation conditions, one-dimensional thermal hydraulic model in U-tubes is established. By the small perturbation theory, the critical pressure drop (CPD) and critical mass flow rate (CMFR) are given and analyzed, which determine the occurrence of reverse flow in U-tubes. The effect of rolling movement on the flow instability in U-tubes is discussed, and the detailed conclusions are drawn as follows:
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J. Hao et al. / Annals of Nuclear Energy 64 (2014) 295–300
(1) The rolling movement has a certain effect on the relationship between pressure drop and mass flow rate. (2) The CPD changes periodically under the rolling movement, and the fluctuation amplitude of CPD decreases with the increase of rolling period. The fluctuation period of CPD is the same as the rolling period under the transverse swing movement, but it is the half of the rolling period under the longitudinal swing movement. The transverse swing movement has larger effect on the fluctuation amplitude of CPD than the longitudinal swing movement. (3) The relationship between the fluctuation amplitude of CPD and rolling amplitude is linear function under the transverse swing movement, and it is square function under the longitudinal swing movement. (4) The transverse swing movement has an obvious effect on the space distribution of reverse flow, but the longitudinal swing movement has little effect on the space distribution of reverse flow.
Acknowledgement This research is supported by the Doctorial Innovation Fund of Naval University of Engineering. References Chen, Z.Y., Hao, J.L., Chen, W.Z., 2012. The development of fast simulation program for marine reactor parameters. Ann. Nucl. Energy 40, 45–52.
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