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Experimental and theoretical study on single-phase natural circulation flow and heat transfer under rolling motion condition
Applied Thermal Engineering 29 (2009) 3160–3168
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Experimental and theoretical study on single-phase natural circulation flow and heat transfer under rolling motion condition Tan Si-chao a,b,1, G.H. Su a,*, Gao Pu-zhen b,1 a
State Key Laboratory of Multiphase Flow in Power Engineering, Department of Nuclear Science and Technology, Xi’an Jiaotong University, 28, Xianning West Road, Shaanxi, Xi’an 710049, China b College of Nuclear Science and Technology, Harbin Engineering University, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 9 February 2008 Accepted 7 April 2009 Available online 14 April 2009 Keywords: Natural circulation Rolling motion Heat transfer Friction coefficient Mathematical model
a b s t r a c t Experimental and theoretical studies of single-phase natural circulation flow and heat transfer under a rolling motion condition are performed. Experiments with and without rolling motions are conducted so the effects of rolling motion on natural circulation flow and heat transfer are obtained. The experimental results show the additional inertia caused by rolling motion easily causes the natural circulation flow to fluctuate. The average mass flow rate of natural circulation decreases with increases in rolling amplitude and frequency. Rolling motion enhances the heat transfer, and the heat transfer coefficient of natural circulation flow increases with the rolling amplitude and frequency. An empirical equation for the heat transfer coefficient under rolling motion is achieved, and a mathematical model is also developed to calculate the natural circulation flow under a rolling motion condition. The calculated results agree well with experimental data. Effects of the rolling motion on natural circulation flow are analyzed using the model. The increase in the flow resistance coefficient is the main reason why the natural circulation capacity decreases under a rolling motion condition. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Natural circulation has extensive application in nuclear power engineering [1], as it does for deep sea reactor research [2]. Nuclear powered ships are rocked by ocean waves that introduce an additional acceleration to the ship reactors [3,4]. The additional inertia makes the primary coolant system flow fluctuate periodically and therefore it is easy for system instability to occur [5,6]. Thus it is important to clarify the effects of such ocean conditions as rolling motion upon natural circulation to ensure reactor safety. In recent years, several studies have focused on the thermal hydraulics of a reactor under ocean conditions [2,4–28]. Ozoe et al. [7], Iyori et al. [8,9] and Kim et al. [10] experimentally studied the natural circulation flow under an incline condition and analyzed the effect of the incline angle. Ishida studied the thermalhydraulic behavior of a marine reactor system [2,11–13] and the results indicated the overlapping of flow oscillation caused by heaving motion and self-excited oscillation resulted in the system being more unstable. Compared with the effects of inclining and heaving motions, those of a rolling motion are more complicated. An inclining motion only means a change in the effective height of the natural cir* Corresponding author. Tel./fax: +86 29 82663401. E-mail addresses: [email protected] (S.-c. Tan), [email protected] (G.H. Su). 1 Tel./fax: +86 451 82569655. 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.04.019
culation flow loop, and a heaving motion only introduces an additional acceleration to the gravity acceleration. However, a rolling motion not only changes the effective height, but also introduces three additional accelerations, namely centripetal, tangential and Coriolis accelerations [3,29]. Murata et al. [14–16], Tan et al. [17–21] and Gao et al. [22] experimentally studied natural circulation flow under a rolling motion condition and their results show the coolant mass flow rate changes periodically with the rolling angle owing to the inertial force caused by the rolling motion. Similar results were obtained in theoretical studies conducted by Fuji [4], Gao et al. [23,24], Su et al. [25], Yang et al. [26,27] and Guo et al. [28]. Because the additional inertia on a test loop varies according to its distance from the rolling axis with different test apparatuses, the effects of rolling motion are also different. Using a test apparatus with two loops, Yang [26] believes the period of the natural calculation flow fluctuation is half the rolling period because of the effect of two circulation loops. However, the experimental results of Ref. [16] show the periods of natural calculation flow fluctuation in both the hot and cold legs are the same as the period of the rolling motion, and the mass flow rate in the core does not oscillate. Using the test apparatus with a single loop, experimental results [17–22] show the period of flow fluctuation is the same as the rolling period. The axis of the rolling plant lays above the top of the loop for Ref. [16], is in the middle of the plant for Refs. [17–21] and lays under the loop for Ref. [22]. Therefore, the effects might be different.
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Nomenclature a A Aw b c d e d cp CR f* g h h hr r h k l w p P Pc Pin Pout Dpg Dpa
coefficient cross sectional area of flow tube, m2 cross sectional area of heat tube, m2 coefficient coefficient coefficient coefficient diameter of flow tube, m specific heat, kJ/kg influence factor of the rolling motion relative rolling frequency gravitational acceleration, m/s2 height of vertical section, m heat transfer coefficient, W/(m2 °C) heat transfer coefficient for rolling motion, W/(m2 °C) average heat transfer coefficient for rolling motion, W/ (m2 °C) local friction resistance coefficient length of horizontal section, m mass flow rate, kg/s pressure, MPa power, kW heat capacity of heated tube, kW power absorbed by the work fluid, kW power loss, kW gravitational pressure drop, MPa additional pressure drop, MPa
Though the previous research results confirm the rolling motion induces a periodic fluid flow fluctuation, the effects of the system parameters on the characteristics of natural circulation flow have not yet been studied in detail. The heat transfer under rolling motion has been studied only by Murata et al. [16]. Their results show the heat transfer in the core is enhanced, which is thought to be caused by internal flow due to the rolling motion. In Murata et al.’s study, however, the flow in the heating section is almost steady and the coolant continuously fluctuates under most ocean conditions. Therefore, the heat transfer under a condition of oscillating flow should be studied in detail. As the coolant flow fluctuates, the frictional and heat transfer characteristics also differ to those of steady flow [30,31] and an in-depth experimental study of the characteristics is required. Thus the heat transfer and flow characteristics of natural circulation under a rolling motion condition are studied in this paper. 2. Test apparatus 2.1. Test loop The experimental apparatus is shown in Fig. 1. The apparatus consists of a pressurizer, coolant pump, condenser, test section and pre-heater. The total height of the loop is 3 m. The working fluid is distilled water. The fluid is heated in the pre-heater until the fluid temperature reaches the required value. The fluid then flows into the test section and is heated continually in the test section. Thus the fluid temperature increases. The hot fluid then flows into the condenser so that it can be cooled. Finally the cooled fluid returns to the preheater. The test section is a stainless steel tube with inner diameter of 16 mm and tube thickness of 1 mm (denoted by £16 1 mm) that is directly heated using a DC power supply, and the working
Dpf t0 t Tin Tout Tw
m mr m m* h hm h*
q qw Dq Dqm Dqh k
x b bmax + Nu Pr Re
frictional pressure drop, MPa rolling period, s time, s fluid temperature of inlet heat section, °C fluid temperature of outlet heat section, °C temperature of the heat wall, °C mass flow velocity, m/s mass flow velocity under rolling motion, m/s average mass flow velocity, m/s relative amplitude of fluctuation rolling angle, rad rolling amplitude, rad relative rolling amplitude density, kg/m3 density of the heat tube, kg/m3 density difference, kg/m3 density difference of vertical section, kg/m3 density difference of horizontal section, kg/m3 Darcy frictional resistance coefficient angular velocity, rad/s angular acceleration, rad/s2 maximum angular acceleration, rad/s2 wave crest point trough point Nusselt number Prandtl number Reynolds number
fluid flows inside the tube. The length of the heated section is 1.2 m and the vertical height between the outlet of the heated section and the rolling axis is also 1.2 m. The pressurizer is a cylindrical vessel with a float level sensor on its side. The system pressure is obtained through filling with nitrogen gas. Nine electrical heating elements are employed to heat the fluid in the pre-heater and the potential difference can be controlled from 0 to 380 V. The highest electric power of the pre-heater is 45 kW. The condenser is a shell-and-tube type heat exchanger. 2.2. Rolling plate The rolling plate is a 2 m 2.5 m rectangular plane with a horizontal axis across the middle as shown in Fig. 1. The rolling axis is the O–O axis and the height between the rolling axis and condenser is 2.2 m. The plate is horizontal and the fluid in the test section flows upward in the normal case. The rolling plate is driven by a crank and rocker mechanism. Different rolling amplitudes can be obtained through changing the length of the rocker and linkage. The rolling plate can be controlled to roll with a certain period by changing the frequency of the electromotor, to simulate a ship at sea. The rolling motion is simulated by a sine wave. The rolling angle can be approximated by
h ¼ hm sinð2pt=t0 Þ
ð1Þ
The angular velocity and angular acceleration are
x ¼ hm ð2p=t0 Þ cosð2pt=t0 Þ
ð2Þ
b ¼ hm ð2p=t0 Þ2 sinð2pt=t 0 Þ
ð3Þ
The ranges of experimental parameters are inlet subcooling of 0–60 °C, amplitude of rolling of 10°, 15° and 20°, period of rolling of 7.5, 10 and 12.5 s and system pressure of 0.1–0.4 MPa.
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Fig. 1. Experimental apparatus.
Under a rolling motion condition, the coolant fluctuates periodically. The fluctuation is close to being a sine wave. The period of fluctuation is the same as the rolling period. The wall temperature of the test section always varies periodically. However, the fluid temperature in the outlet of the test section does not vary so widely, as shown in Fig. 2. Although the system pressure is the same as that in the non-rolling case, the outlet pressure of the test section decreases if there is rolling, and both system pressure and outlet pressure of the test section do not fluctuate obviously. If the rolling period is short, the outlet temperature of the working fluid of the test section changes so slightly that it can be considered as a constant, as shown in Fig. 3. Compared with the non-rolling case, the relative natural circulation flow velocity (RNCFV) is defined as the ratio of the flow
velocity in the rolling case to that in the non-rolling case with the same experimental conditions. The effects of rolling amplitude on average RNCFV are shown in Fig. 4. The average RNCFV decreases with increasing rolling amplitude. The effects of the rolling period on average RNCFV are shown in Fig. 5, with the average RNCFV increasing as the rolling period increases. Figs. 4 and 5 show the RNCFV is less than 1.0, and thus it is possible to decrease the natural circulation flow velocity under a rolling motion. The reason for this will be discussed in Section 6.1. The effects of rolling amplitude and period on the amplitude of the RNCFV are shown in Figs. 6 and 7, where ‘‘+” and ‘‘” refer to the wave crest and trough of the RNCFV. The amplitude of the RNCFV increases as rolling amplitude and frequency increase. Furthermore, the amplitude of the RNCFV decreases with increasing inlet temperature of the test section. The reason for this will be discussed in Sections 6.1 and 6.3.
Fig. 2. Wall temperature and outlet temperature under flow fluctuation condition.
Fig. 3. Wall temperature and outlet temperature in short rolling period case (7.5 s).
3. Phenomenon
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Fig. 4. Effects of the amplitude of rolling motion on RNCFV.
Fig. 7. Effect of rolling period on amplitude of RNCFV.
4. Heat transfer characteristics 4.1. Comparison of Nusselt numbers for rolling and non-rolling cases Heat transfer characteristics of the natural circulation flow are also influenced by rolling motion. To express the influence, the relr =h is hereby employed and is deative heat transfer coefficient h r under a fined as the ratio of average heat transfer coefficient h rolling motion condition to that (h) under a non-rolling condition in the same experimental conditions, as shown in Table 1. The wall temperature varies periodically and thus the influence of the heat capacity (Pc) of the heat tube cannot be neglected. The experimental data under both rolling and non-rolling conditions fit the heat balance equation very well with an error less than ±5%, as seen in Fig. 8. Pcal is calculated using Eq. (4), and Tout and w are the average outlet temperature and mass flow rate for one rolling period in the rolling case.
Pcal ¼ cp wðT out T in Þ Fig. 5. Effects of the period of rolling motion on RNCFV.
ð4Þ
The heat transfer coefficient under rolling motion is calculated using
hr ¼ ðP P c Þ=pdlðT w TÞ
ð5Þ
where
Pc ¼ qw cpw V w
dT w dt
ð6Þ
r and m r are the heat transfer coefficient and mass flow rate averh aged over one period under the rolling motion condition.
Table 1 Relative heat transfer coefficient.
Fig. 6. Effect of rolling amplitude on amplitude of RNCFV.
t0 (s)
mr/m
hr/h
12.5
0.884 0.875 0.928
1.066 0.987 0.964
10
0.765 0.775 0.818
1.228 1.159 1.103
7.5
0.680 0.666 0.688
1.815 1.880 1.776
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6
120
non-rolling rolling +5%
5
100
Nuc
P / kW
-5% 4
80
+15%
60
3
-15% 40 40
2
2
3
4
5
6
r ) is less than 1.0 Though the relative heat transfer coefficient (h when the rolling period is 12.5 s, considering the average flow r ) under rolling motion is less than that for the non-rollvelocity (m ing case, the heat transfer coefficient with a 12.5 s rolling period is larger than that in the non-rolling case with the same flow velocity. Therefore, the heat transfer is enhanced under rolling motion. Table 1 also shows the relative heat transfer coefficient increases with increasing rolling frequency, especially when the rolling period is 7.5 s. The heat transfer is enhanced under rolling motion due to the coolant fluctuation caused by rolling motion. This was also found by Murata et al. [16]. Summarizing the above discussion, it has been found rolling motion results in coolant fluctuation while the heat transfer is enhanced. With increasing rolling amplitude and rolling frequency, the amplitude of coolant fluctuation increases. Thus the heat transfer coefficient increases. Although the heat transfer increases under rolling motion, the present empirical equations for the heat transfer coefficient are based on steady flow. Thus it is necessary to study how to apply equations in the case of rolling motion. It is known that the overall heat transfer phenomenon may be characterized by the Nusselt number, and the behavior of heat transfer for single phase flow may be characterized by
ð7Þ
If one can obtain the values of constants c, a and b, Eq. (7) is suitable for calculating the heat transfer. Almost all previous research, experimental and theoretical, has used the above strategy. Dittus and Boelter [32] developed an empirical correlation (Eq. (8)), which is the currently used formula. Fig. 9 shows the comparison between present data for the non-rolling case and Eq. (8), and the error is less than ±15%.
Nu ¼ 0:023Re Pr
0:4
100
120
Fig. 9. Comparison between experimental and calculated Nusselt numbers (nonrolling case).
Fig. 8. Comparison between experimental and calculated heat power.
0:8
80
Nue
Pcal / kW
Nu ¼ f ðRe; PrÞ ¼ cRea Pr b
60
ð8Þ
Fig. 10 compares experimental data for the rolling case (10°) and results calculated using Eq. (8). When the rolling motion is slight (12.5 s), the results calculated using Eq. (8) are mainly in accordance with experimental data and the error is less than 20%. In an intense rolling case (7.5 s), the difference between the calculated and experimental results is large. Therefore, Eq. (8) is not suitable for calculating the Nusselt number for intense rolling motion.
4.2. Empirical correlation of the heat transfer coefficient under the rolling motion condition There are two aspects to heat transfer under a rolling motion condition: one relating to natural circulation and the other to rolling motion. The influences of rolling motion on heat transfer characteristics have two different trends. Under a rolling motion condition, the average flow velocity decreases and thus heat transfer weakens. However, the rolling motion can also cause coolant fluctuation, which enhances the heat transfer. The parameters of rolling motion and coolant fluctuation must be considered during the calculation of the Nusselt number in the rolling case. Therefore, three nondimensional numbers (h*, f* and m*) are defined to employ the influence of rolling motion and fluctuation.
h ¼ hm =10 f ¼ 10=t 0
ð9Þ ð10Þ
m ¼ ðmþ m Þ=m
ð11Þ
Based on the above, Eq. (7) is rewritten as
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ cRea Pr b hc f d me
ð12Þ
The fluid is only water and the temperature of the fluid does not change as much as the wall temperature does, so the index b in Eq. (12) is presumed as 0.4 from Eq. (8). On the basis of experimental data, one can obtain the empirical equation using a least square technique based on Eq. (12):
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 1:031Re0:44 Pr0:4 h0:57 f 1:18 m0:21
ð13Þ
The relative error of Eq. (13) is less than ±30%, as shown in Fig. 11. Considering the exponents f* and h* in Eq. (13), 1.18 is about double 0.57. Thus Eq. (13) can be rewritten as Eq. (14) and the error is again less than ±30%.
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 0:264Re0:44 Pr0:4
hm t 20
!0:59
m0:21
ð14Þ
From Eq. (3), the maximum angular acceleration of rolling motion can be expressed as
bmax ¼ 4p2 ðhm =t20 Þ
ð15Þ
Therefore, Eq. (13) can be transformed as 0:21 Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 0:0302Re0:44 Pr0:4 b0:59 max m
ð16Þ
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110
calculation experiment
100 90
Nu r
80 70 60 50 40
12.5s
7.5s
30 20 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
relative time (t/t 0 ) Fig. 10. Comparison between experimental and calculated Nusselt numbers (rolling of 10°).
Eq. (16) clearly shows the maximum angular acceleration of rolling motion bmax may strongly influence the heat transfer under a rolling motion condition.
ZZZ
ZZZ s0
½f a00 x ðx r 0 Þ b r 0 2x V 0 qds0
þ tA0 pn dA
0
ð18Þ
a00
5. Mathematical model for natural circulation flow Mass, momentum and energy conservation equations must be developed to describe the natural circulation flow. 5.1. Mass conservation equation
where f is the mass force and here f = g; is the movement accel0 0 eration of the relative coordinates and here a00 ¼ 0; DDtV is the relative 0 0 acceleration; x ðx r Þ is the centripetal acceleration; b r is 0 the tangent acceleration; and 2x V is the Coriolis acceleration. The later three terms are additional accelerations caused by the rolling motion. Eq. (18) may be re-written as
X
Because the natural circulation loop is closed, the mass conservation equation is described as
qAm ¼ C
D0 V 0 qds0 ¼ s0 Dt
ð17Þ
qi
X X dmi X Dpg þ Dpf þ Dpa ¼ dt
The gravitational pressure drop under rolling motion Dpg is calculated by
Dpg ¼ Dqm hm cos h þ Dqh hh sin h 5.2. Momentum conservation equation The momentum conservation equation in non-inertial coordinates is expressed as
Nuc
200
+30%
150
-30% 100
50
50
100
150
200
250
Nue Fig. 11. Comparison between experimental and calculated values for Nur.
ð20Þ
It is valuable to note that the density difference of the horizontal section should be considered. For the non-rolling case, h = 0 . The frictional pressure drop under rolling motion Dpf is calculated by
Dpf ¼ C R
250
ð19Þ
X
k
L qm2 X qv 2 þ k De 2 2
ð21Þ
where the frictional resistance coefficients k and k are calculated by methods in the literature [33]. CR is the influence factor of the rolling motion. CR = 1.0 in the non-rolling case and is first presumed as 1.0 in the rolling case. The force analysis for the natural circulation under rolling motion is shown in Fig. 12. Here, the symbol a is employed to express the relative angle between a certain differential element of the loop and the relative horizontal axis (O–O axis). For the vertical part of the natural circulation loop the angular acceleration in the flow direction is written as br cos a and the centripetal acceleration in the flow direction is described as x2r sin a, i.e. x2z; and for the horizontal part, the angular acceleration in flow direction is expressed as br sin a and the centripetal acceleration in the flow direction is described as x2r cos a, i.e. x2y. For the different tubes of natural circulation, the angular acceleration in the flow direction is bli, where li is the distance between the tube and rolling axis.
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5.4. Modification terms Considering the experimental data, the average flow velocity in the rolling motion case is lower than that in the non-rolling case (Figs. 4 and 5). However, the computational results according to the above model show the average flow velocity of the natural circulation loop in the rolling case is the same as that in the non-rolling case. This may be because CR = 1.0 is assumed. This assumption results in the frictional resistance coefficient being calculated by the normal method without consideration of rolling motion. In other words, in the present model, the frictional resistance coefficient in the rolling case is presumed to be the same as that in the non-rolling case. The rolling motion and coolant flow fluctuation influence not only the heat transfer coefficient but also the frictional resistance coefficient. Therefore, the frictional resistance coefficient should be modified. For a flow fluctuation, there are three important parameters to describe the flow velocity, which are the maximum (m+), minimum Þ. With different values for CR, (m) and average flow velocities ðm . Using experimental data, one can obtain different values for m the values of CR for different rolling cases are obtained and shown in Table 2. An empirical correlation (Eq. (28)) is then obtained with an error less than ±4%, as shown in Fig. 13. Eq. (28) shows the frictional resistance coefficient for rolling motion is larger than that in the non-rolling case. The theoretically calculated RNCFV with the modification agrees well with the experimental data, as shown in Fig. 14 and Table 3.
C R ¼ 0:8942 þ 5:17265ðhm =t20 Þ ¼ 0:8942 þ 0:131bmax
Fig. 12. Force analysis for rolling case.
By integrating the added relative acceleration over each differential control volume, one can obtain the additional pressure drop caused by angular acceleration.
X
bli lj qi ¼ b
X
li lj qi
ð22Þ
The additional pressure drop caused by centripetal acceleration is expressed as
XZ
x2 zi qi dz þ
XZ
x2 yi qi dy ¼ x2
X Z
zi qi dz þ
XZ
yi qi dy
ð23Þ The Coriolis acceleration is perpendicular to the flow direction and thus cannot cause any additional pressure drop. 5.3. Energy conservation equation The energy conservation equation under rolling motion is written as
Pc ¼ P P in Pout
ð24Þ
where Pc is the heat capacity of the heated tube, P is the input power, Pin is the power absorbed by the work fluid, and Pout is power loss with Pout = 0 except for the heated section. They are expressed as
dT w dPc ¼ qw cpw Aw dL dt dPin ¼ ain ðT w TÞpDin dL dPout ¼ ae ðT w T 1 ÞpDout dL
ð25Þ ð26Þ
Eq. (28) shows the maximum angular acceleration bmax strongly influences the flow friction resistance coefficient under rolling motion. This result is very similar to the influence of the maximum angular acceleration on the heat transfer. Therefore, the maximum angular acceleration is the most important parameter influencing the characteristics of natural circulation flow under rolling motion. In fact, the heat transfer is not increased by rolling motion directly, but is enhanced by the flow fluctuation. If the rolling motion cannot vary the flow velocity, it may not influence the heat transfer. Such conclusions were also found by Pendyala et al. [30,31]. 6. Discussion 6.1. Effect of rolling motion on flow velocity The calculated average flow velocities under different rolling motion conditions are the same as those in the non-rolling case and the average RNCFV is 1.0, if the influence factor of the rolling motion CR is unchanged. However, the calculated average RNCFV agrees well with the experimental RNCFV with a modified CR. The average flow velocity under rolling motion and average RNCFV decrease as a result of the flow frictional coefficient being modified. Thus the increase in flow friction is the main factor in decreasTable 2 Coefficient CR in different rolling case. hm (°)
t0 (s)
CR
10
12.5 10 7.5
1.27 1.44 1.87
15
12.5 10 7.5
1.36 1.66 2.31
20
12.5 10 7.5
1.5 1.86 2.7
ð27Þ
where ain is calculated using Eq. (13) and ae is calculated using methods in the literature [32]. Thus a set of equations has been developed based on the mass, momentum and energy conservation equations and the available constitutive equations, and the equations can be solved by the Runge–Kutta method.
ð28Þ
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2.8 2.6 2.4
CR-cal
2.2 2.0 1.8 1.6 1.4 1.2 1.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
CR-exp Fig. 13. Comparison between experimental and calculated values for CR.
calculation without modification calculation with modification experimental data
1.5 1.4 1.3
v r /v
1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0
10
20
30
40
t /s
creases under rolling motion, because of the increase in the frictional pressure drop. The fluctuation amplitude increases with increasing rolling amplitude and frequency. This is because (a) the additional acceleration and additional pressure drops increase and (b) the degree of the incline increases with increasing rolling amplitude, the effect of horizontal parts of the natural circulation loop increases, and thus the variation of the driving head increases. With increasing rolling amplitude and frequency, the amplitude of fluctuation increases, and thus the crest point of flow fluctuation increases and the trough point of flow fluctuation decreases. However, the average flow velocity decreases, so the trough point decreases greatly and the crest point increases only a little. Hence, the increment in the wave crest points of the relative flow velocity is smaller than the decrease in amplitude of the trough points and the influence of rolling motion is greater on the trough point of the flow fluctuation than on the wave crest point, as shown in Figs. 6 and 7. 6.2. Effects of angular and centripetal accelerations From Eqs. (3) and (22), in one rolling period, the direction of angular acceleration changes with time and there is a consistent direction of flow around the loop at any given time because the rolling axis is along the middle of the loop. Thus the effects of angular acceleration on different parts accumulate. From Eqs. (2) and (23), the direction of centripetal acceleration does not change with time. Because the rolling axis is along the middle of the loop and the loop is symmetrical, the effects of centripetal acceleration on different parts mostly counteract one another. Therefore, angular acceleration is the primary influence of rolling motion on the natural circulation within our experimental system and such a conclusion can be proved by Eqs. (16) and (28). Furthermore, Eqs. (13) and (28) cannot be extended to calculate the heat transfer coefficient and flow frictional coefficient for other experimental systems. However, the equations indicate that natural circulation flow is influenced by the acceleration induced by the rolling motion, and that heat transfer is influenced by the additional flow. 6.3. Effect of inlet subcooling of the test section
Fig. 14. Comparison of calculated and experimental data.
ing the average RNCFV. CR increases with increasing rolling amplitude and frequency. Thus the average RNCFV decreases with increasing rolling amplitude and frequency. Furthermore, the increase in flow friction is the reason why the outlet pressure de-
The amplitude of flow fluctuation decreases with decreasing inlet subcooling of the test section as shown in Figs. 6 and 7. If the inlet subcooling decreases, then the density difference between hot and cold legs increases and the average density of the whole loop decreases, so the amplitude of the additional pressure drop decreases and the effect of the driving head increases. Therefore,
Table 3 Comparisons between the calculated and experimental results. Rolling amplitude/period
me =m0
mc =m0
mþe =m0
mþc =m0
me =m0
mc =m0
20°/12.5 s
0.897 0.891 0.937
0.882 0.889 0.897
1.315 1.306 1.298
1.393 1.370 1.352
0.345 0.426 0.456
0.371 0.409 0.443
20°/10 s
0.785 0.792 0.838 0.856
0.789 0.795 0.802 0.827
1.280 1.262 1.262 1.223
1.428 1.397 1.371 1.275
0.214 0.262 0.319 0.396
0.149 0.193 0.233 0.269
10°/10 s
0.884 0.908 0.918
0.892 0.909 0.926
1.203 1.242 1.154
1.201 1.186 1.176
0.555 0.680 0.665
0.583 0.633 0.676
10°/7.5 s
0.743 0.809 0.830
0.786 0.798 0.810
1.055 1.113 1.112
1.199 1.168 1.144
0.388 0.495 0.576
0.375 0.430 0.477
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under the same experimental conditions, the amplitude of fluctuation decreases with increasing inlet temperature.
[9]
6.4. Effect of system pressure [10]
The liquid coolant is distilled water in the experiments. The water density almost does not vary in the range of experimental pressure. Thus the system pressure does not influence the coolant fluctuation.
[11] [12] [13]
7. Conclusions [14]
This paper experimentally and theoretically studies the singlephase natural circulation flow and heat transfer under a rolling motion condition. The rolling motion makes the fluid flow fluctuate periodically. The period of fluctuation is the same as the rolling period, and the fluctuation is close to being a sine wave. The fluctuation amplitude increases with increases in the rolling amplitude, rolling frequency and inlet subcooling. The primary influencing factors of the rolling motion are additional accelerations, among which angular acceleration is paramount. The increase in the friction resistance coefficient caused by flow fluctuation is the main reason why the average RNCFV decreases under rolling motion. The rolling motion is beneficial to heat transfer. The heat transfer coefficient in the rolling case is higher than that in the non-rolling case. Eq. (13) is suitable for calculating the heat transfer coefficient for natural circulation under rolling motion in the parameter ranges of the present study. A mathematical model was developed to simulate the natural circulation flow and heat transfer under rolling motion with an acceptable relative error.
[15]
[16]
[17]
[18]
[19]
[20]
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Experimental and theoretical study on single-phase natural circulation flow and heat transfer under rolling motion condition Tan Si-chao a,b,1, G.H. Su a,*, Gao Pu-zhen b,1 a
State Key Laboratory of Multiphase Flow in Power Engineering, Department of Nuclear Science and Technology, Xi’an Jiaotong University, 28, Xianning West Road, Shaanxi, Xi’an 710049, China b College of Nuclear Science and Technology, Harbin Engineering University, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 9 February 2008 Accepted 7 April 2009 Available online 14 April 2009 Keywords: Natural circulation Rolling motion Heat transfer Friction coefficient Mathematical model
a b s t r a c t Experimental and theoretical studies of single-phase natural circulation flow and heat transfer under a rolling motion condition are performed. Experiments with and without rolling motions are conducted so the effects of rolling motion on natural circulation flow and heat transfer are obtained. The experimental results show the additional inertia caused by rolling motion easily causes the natural circulation flow to fluctuate. The average mass flow rate of natural circulation decreases with increases in rolling amplitude and frequency. Rolling motion enhances the heat transfer, and the heat transfer coefficient of natural circulation flow increases with the rolling amplitude and frequency. An empirical equation for the heat transfer coefficient under rolling motion is achieved, and a mathematical model is also developed to calculate the natural circulation flow under a rolling motion condition. The calculated results agree well with experimental data. Effects of the rolling motion on natural circulation flow are analyzed using the model. The increase in the flow resistance coefficient is the main reason why the natural circulation capacity decreases under a rolling motion condition. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Natural circulation has extensive application in nuclear power engineering [1], as it does for deep sea reactor research [2]. Nuclear powered ships are rocked by ocean waves that introduce an additional acceleration to the ship reactors [3,4]. The additional inertia makes the primary coolant system flow fluctuate periodically and therefore it is easy for system instability to occur [5,6]. Thus it is important to clarify the effects of such ocean conditions as rolling motion upon natural circulation to ensure reactor safety. In recent years, several studies have focused on the thermal hydraulics of a reactor under ocean conditions [2,4–28]. Ozoe et al. [7], Iyori et al. [8,9] and Kim et al. [10] experimentally studied the natural circulation flow under an incline condition and analyzed the effect of the incline angle. Ishida studied the thermalhydraulic behavior of a marine reactor system [2,11–13] and the results indicated the overlapping of flow oscillation caused by heaving motion and self-excited oscillation resulted in the system being more unstable. Compared with the effects of inclining and heaving motions, those of a rolling motion are more complicated. An inclining motion only means a change in the effective height of the natural cir* Corresponding author. Tel./fax: +86 29 82663401. E-mail addresses: [email protected] (S.-c. Tan), [email protected] (G.H. Su). 1 Tel./fax: +86 451 82569655. 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.04.019
culation flow loop, and a heaving motion only introduces an additional acceleration to the gravity acceleration. However, a rolling motion not only changes the effective height, but also introduces three additional accelerations, namely centripetal, tangential and Coriolis accelerations [3,29]. Murata et al. [14–16], Tan et al. [17–21] and Gao et al. [22] experimentally studied natural circulation flow under a rolling motion condition and their results show the coolant mass flow rate changes periodically with the rolling angle owing to the inertial force caused by the rolling motion. Similar results were obtained in theoretical studies conducted by Fuji [4], Gao et al. [23,24], Su et al. [25], Yang et al. [26,27] and Guo et al. [28]. Because the additional inertia on a test loop varies according to its distance from the rolling axis with different test apparatuses, the effects of rolling motion are also different. Using a test apparatus with two loops, Yang [26] believes the period of the natural calculation flow fluctuation is half the rolling period because of the effect of two circulation loops. However, the experimental results of Ref. [16] show the periods of natural calculation flow fluctuation in both the hot and cold legs are the same as the period of the rolling motion, and the mass flow rate in the core does not oscillate. Using the test apparatus with a single loop, experimental results [17–22] show the period of flow fluctuation is the same as the rolling period. The axis of the rolling plant lays above the top of the loop for Ref. [16], is in the middle of the plant for Refs. [17–21] and lays under the loop for Ref. [22]. Therefore, the effects might be different.
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Nomenclature a A Aw b c d e d cp CR f* g h h hr r h k l w p P Pc Pin Pout Dpg Dpa
coefficient cross sectional area of flow tube, m2 cross sectional area of heat tube, m2 coefficient coefficient coefficient coefficient diameter of flow tube, m specific heat, kJ/kg influence factor of the rolling motion relative rolling frequency gravitational acceleration, m/s2 height of vertical section, m heat transfer coefficient, W/(m2 °C) heat transfer coefficient for rolling motion, W/(m2 °C) average heat transfer coefficient for rolling motion, W/ (m2 °C) local friction resistance coefficient length of horizontal section, m mass flow rate, kg/s pressure, MPa power, kW heat capacity of heated tube, kW power absorbed by the work fluid, kW power loss, kW gravitational pressure drop, MPa additional pressure drop, MPa
Though the previous research results confirm the rolling motion induces a periodic fluid flow fluctuation, the effects of the system parameters on the characteristics of natural circulation flow have not yet been studied in detail. The heat transfer under rolling motion has been studied only by Murata et al. [16]. Their results show the heat transfer in the core is enhanced, which is thought to be caused by internal flow due to the rolling motion. In Murata et al.’s study, however, the flow in the heating section is almost steady and the coolant continuously fluctuates under most ocean conditions. Therefore, the heat transfer under a condition of oscillating flow should be studied in detail. As the coolant flow fluctuates, the frictional and heat transfer characteristics also differ to those of steady flow [30,31] and an in-depth experimental study of the characteristics is required. Thus the heat transfer and flow characteristics of natural circulation under a rolling motion condition are studied in this paper. 2. Test apparatus 2.1. Test loop The experimental apparatus is shown in Fig. 1. The apparatus consists of a pressurizer, coolant pump, condenser, test section and pre-heater. The total height of the loop is 3 m. The working fluid is distilled water. The fluid is heated in the pre-heater until the fluid temperature reaches the required value. The fluid then flows into the test section and is heated continually in the test section. Thus the fluid temperature increases. The hot fluid then flows into the condenser so that it can be cooled. Finally the cooled fluid returns to the preheater. The test section is a stainless steel tube with inner diameter of 16 mm and tube thickness of 1 mm (denoted by £16 1 mm) that is directly heated using a DC power supply, and the working
Dpf t0 t Tin Tout Tw
m mr m m* h hm h*
q qw Dq Dqm Dqh k
x b bmax + Nu Pr Re
frictional pressure drop, MPa rolling period, s time, s fluid temperature of inlet heat section, °C fluid temperature of outlet heat section, °C temperature of the heat wall, °C mass flow velocity, m/s mass flow velocity under rolling motion, m/s average mass flow velocity, m/s relative amplitude of fluctuation rolling angle, rad rolling amplitude, rad relative rolling amplitude density, kg/m3 density of the heat tube, kg/m3 density difference, kg/m3 density difference of vertical section, kg/m3 density difference of horizontal section, kg/m3 Darcy frictional resistance coefficient angular velocity, rad/s angular acceleration, rad/s2 maximum angular acceleration, rad/s2 wave crest point trough point Nusselt number Prandtl number Reynolds number
fluid flows inside the tube. The length of the heated section is 1.2 m and the vertical height between the outlet of the heated section and the rolling axis is also 1.2 m. The pressurizer is a cylindrical vessel with a float level sensor on its side. The system pressure is obtained through filling with nitrogen gas. Nine electrical heating elements are employed to heat the fluid in the pre-heater and the potential difference can be controlled from 0 to 380 V. The highest electric power of the pre-heater is 45 kW. The condenser is a shell-and-tube type heat exchanger. 2.2. Rolling plate The rolling plate is a 2 m 2.5 m rectangular plane with a horizontal axis across the middle as shown in Fig. 1. The rolling axis is the O–O axis and the height between the rolling axis and condenser is 2.2 m. The plate is horizontal and the fluid in the test section flows upward in the normal case. The rolling plate is driven by a crank and rocker mechanism. Different rolling amplitudes can be obtained through changing the length of the rocker and linkage. The rolling plate can be controlled to roll with a certain period by changing the frequency of the electromotor, to simulate a ship at sea. The rolling motion is simulated by a sine wave. The rolling angle can be approximated by
h ¼ hm sinð2pt=t0 Þ
ð1Þ
The angular velocity and angular acceleration are
x ¼ hm ð2p=t0 Þ cosð2pt=t0 Þ
ð2Þ
b ¼ hm ð2p=t0 Þ2 sinð2pt=t 0 Þ
ð3Þ
The ranges of experimental parameters are inlet subcooling of 0–60 °C, amplitude of rolling of 10°, 15° and 20°, period of rolling of 7.5, 10 and 12.5 s and system pressure of 0.1–0.4 MPa.
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Fig. 1. Experimental apparatus.
Under a rolling motion condition, the coolant fluctuates periodically. The fluctuation is close to being a sine wave. The period of fluctuation is the same as the rolling period. The wall temperature of the test section always varies periodically. However, the fluid temperature in the outlet of the test section does not vary so widely, as shown in Fig. 2. Although the system pressure is the same as that in the non-rolling case, the outlet pressure of the test section decreases if there is rolling, and both system pressure and outlet pressure of the test section do not fluctuate obviously. If the rolling period is short, the outlet temperature of the working fluid of the test section changes so slightly that it can be considered as a constant, as shown in Fig. 3. Compared with the non-rolling case, the relative natural circulation flow velocity (RNCFV) is defined as the ratio of the flow
velocity in the rolling case to that in the non-rolling case with the same experimental conditions. The effects of rolling amplitude on average RNCFV are shown in Fig. 4. The average RNCFV decreases with increasing rolling amplitude. The effects of the rolling period on average RNCFV are shown in Fig. 5, with the average RNCFV increasing as the rolling period increases. Figs. 4 and 5 show the RNCFV is less than 1.0, and thus it is possible to decrease the natural circulation flow velocity under a rolling motion. The reason for this will be discussed in Section 6.1. The effects of rolling amplitude and period on the amplitude of the RNCFV are shown in Figs. 6 and 7, where ‘‘+” and ‘‘” refer to the wave crest and trough of the RNCFV. The amplitude of the RNCFV increases as rolling amplitude and frequency increase. Furthermore, the amplitude of the RNCFV decreases with increasing inlet temperature of the test section. The reason for this will be discussed in Sections 6.1 and 6.3.
Fig. 2. Wall temperature and outlet temperature under flow fluctuation condition.
Fig. 3. Wall temperature and outlet temperature in short rolling period case (7.5 s).
3. Phenomenon
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Fig. 4. Effects of the amplitude of rolling motion on RNCFV.
Fig. 7. Effect of rolling period on amplitude of RNCFV.
4. Heat transfer characteristics 4.1. Comparison of Nusselt numbers for rolling and non-rolling cases Heat transfer characteristics of the natural circulation flow are also influenced by rolling motion. To express the influence, the relr =h is hereby employed and is deative heat transfer coefficient h r under a fined as the ratio of average heat transfer coefficient h rolling motion condition to that (h) under a non-rolling condition in the same experimental conditions, as shown in Table 1. The wall temperature varies periodically and thus the influence of the heat capacity (Pc) of the heat tube cannot be neglected. The experimental data under both rolling and non-rolling conditions fit the heat balance equation very well with an error less than ±5%, as seen in Fig. 8. Pcal is calculated using Eq. (4), and Tout and w are the average outlet temperature and mass flow rate for one rolling period in the rolling case.
Pcal ¼ cp wðT out T in Þ Fig. 5. Effects of the period of rolling motion on RNCFV.
ð4Þ
The heat transfer coefficient under rolling motion is calculated using
hr ¼ ðP P c Þ=pdlðT w TÞ
ð5Þ
where
Pc ¼ qw cpw V w
dT w dt
ð6Þ
r and m r are the heat transfer coefficient and mass flow rate averh aged over one period under the rolling motion condition.
Table 1 Relative heat transfer coefficient.
Fig. 6. Effect of rolling amplitude on amplitude of RNCFV.
t0 (s)
mr/m
hr/h
12.5
0.884 0.875 0.928
1.066 0.987 0.964
10
0.765 0.775 0.818
1.228 1.159 1.103
7.5
0.680 0.666 0.688
1.815 1.880 1.776
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6
120
non-rolling rolling +5%
5
100
Nuc
P / kW
-5% 4
80
+15%
60
3
-15% 40 40
2
2
3
4
5
6
r ) is less than 1.0 Though the relative heat transfer coefficient (h when the rolling period is 12.5 s, considering the average flow r ) under rolling motion is less than that for the non-rollvelocity (m ing case, the heat transfer coefficient with a 12.5 s rolling period is larger than that in the non-rolling case with the same flow velocity. Therefore, the heat transfer is enhanced under rolling motion. Table 1 also shows the relative heat transfer coefficient increases with increasing rolling frequency, especially when the rolling period is 7.5 s. The heat transfer is enhanced under rolling motion due to the coolant fluctuation caused by rolling motion. This was also found by Murata et al. [16]. Summarizing the above discussion, it has been found rolling motion results in coolant fluctuation while the heat transfer is enhanced. With increasing rolling amplitude and rolling frequency, the amplitude of coolant fluctuation increases. Thus the heat transfer coefficient increases. Although the heat transfer increases under rolling motion, the present empirical equations for the heat transfer coefficient are based on steady flow. Thus it is necessary to study how to apply equations in the case of rolling motion. It is known that the overall heat transfer phenomenon may be characterized by the Nusselt number, and the behavior of heat transfer for single phase flow may be characterized by
ð7Þ
If one can obtain the values of constants c, a and b, Eq. (7) is suitable for calculating the heat transfer. Almost all previous research, experimental and theoretical, has used the above strategy. Dittus and Boelter [32] developed an empirical correlation (Eq. (8)), which is the currently used formula. Fig. 9 shows the comparison between present data for the non-rolling case and Eq. (8), and the error is less than ±15%.
Nu ¼ 0:023Re Pr
0:4
100
120
Fig. 9. Comparison between experimental and calculated Nusselt numbers (nonrolling case).
Fig. 8. Comparison between experimental and calculated heat power.
0:8
80
Nue
Pcal / kW
Nu ¼ f ðRe; PrÞ ¼ cRea Pr b
60
ð8Þ
Fig. 10 compares experimental data for the rolling case (10°) and results calculated using Eq. (8). When the rolling motion is slight (12.5 s), the results calculated using Eq. (8) are mainly in accordance with experimental data and the error is less than 20%. In an intense rolling case (7.5 s), the difference between the calculated and experimental results is large. Therefore, Eq. (8) is not suitable for calculating the Nusselt number for intense rolling motion.
4.2. Empirical correlation of the heat transfer coefficient under the rolling motion condition There are two aspects to heat transfer under a rolling motion condition: one relating to natural circulation and the other to rolling motion. The influences of rolling motion on heat transfer characteristics have two different trends. Under a rolling motion condition, the average flow velocity decreases and thus heat transfer weakens. However, the rolling motion can also cause coolant fluctuation, which enhances the heat transfer. The parameters of rolling motion and coolant fluctuation must be considered during the calculation of the Nusselt number in the rolling case. Therefore, three nondimensional numbers (h*, f* and m*) are defined to employ the influence of rolling motion and fluctuation.
h ¼ hm =10 f ¼ 10=t 0
ð9Þ ð10Þ
m ¼ ðmþ m Þ=m
ð11Þ
Based on the above, Eq. (7) is rewritten as
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ cRea Pr b hc f d me
ð12Þ
The fluid is only water and the temperature of the fluid does not change as much as the wall temperature does, so the index b in Eq. (12) is presumed as 0.4 from Eq. (8). On the basis of experimental data, one can obtain the empirical equation using a least square technique based on Eq. (12):
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 1:031Re0:44 Pr0:4 h0:57 f 1:18 m0:21
ð13Þ
The relative error of Eq. (13) is less than ±30%, as shown in Fig. 11. Considering the exponents f* and h* in Eq. (13), 1.18 is about double 0.57. Thus Eq. (13) can be rewritten as Eq. (14) and the error is again less than ±30%.
Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 0:264Re0:44 Pr0:4
hm t 20
!0:59
m0:21
ð14Þ
From Eq. (3), the maximum angular acceleration of rolling motion can be expressed as
bmax ¼ 4p2 ðhm =t20 Þ
ð15Þ
Therefore, Eq. (13) can be transformed as 0:21 Nu ¼ f ðRe; Pr; h ; f ; m Þ ¼ 0:0302Re0:44 Pr0:4 b0:59 max m
ð16Þ
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110
calculation experiment
100 90
Nu r
80 70 60 50 40
12.5s
7.5s
30 20 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
relative time (t/t 0 ) Fig. 10. Comparison between experimental and calculated Nusselt numbers (rolling of 10°).
Eq. (16) clearly shows the maximum angular acceleration of rolling motion bmax may strongly influence the heat transfer under a rolling motion condition.
ZZZ
ZZZ s0
½f a00 x ðx r 0 Þ b r 0 2x V 0 qds0
þ tA0 pn dA
0
ð18Þ
a00
5. Mathematical model for natural circulation flow Mass, momentum and energy conservation equations must be developed to describe the natural circulation flow. 5.1. Mass conservation equation
where f is the mass force and here f = g; is the movement accel0 0 eration of the relative coordinates and here a00 ¼ 0; DDtV is the relative 0 0 acceleration; x ðx r Þ is the centripetal acceleration; b r is 0 the tangent acceleration; and 2x V is the Coriolis acceleration. The later three terms are additional accelerations caused by the rolling motion. Eq. (18) may be re-written as
X
Because the natural circulation loop is closed, the mass conservation equation is described as
qAm ¼ C
D0 V 0 qds0 ¼ s0 Dt
ð17Þ
qi
X X dmi X Dpg þ Dpf þ Dpa ¼ dt
The gravitational pressure drop under rolling motion Dpg is calculated by
Dpg ¼ Dqm hm cos h þ Dqh hh sin h 5.2. Momentum conservation equation The momentum conservation equation in non-inertial coordinates is expressed as
Nuc
200
+30%
150
-30% 100
50
50
100
150
200
250
Nue Fig. 11. Comparison between experimental and calculated values for Nur.
ð20Þ
It is valuable to note that the density difference of the horizontal section should be considered. For the non-rolling case, h = 0 . The frictional pressure drop under rolling motion Dpf is calculated by
Dpf ¼ C R
250
ð19Þ
X
k
L qm2 X qv 2 þ k De 2 2
ð21Þ
where the frictional resistance coefficients k and k are calculated by methods in the literature [33]. CR is the influence factor of the rolling motion. CR = 1.0 in the non-rolling case and is first presumed as 1.0 in the rolling case. The force analysis for the natural circulation under rolling motion is shown in Fig. 12. Here, the symbol a is employed to express the relative angle between a certain differential element of the loop and the relative horizontal axis (O–O axis). For the vertical part of the natural circulation loop the angular acceleration in the flow direction is written as br cos a and the centripetal acceleration in the flow direction is described as x2r sin a, i.e. x2z; and for the horizontal part, the angular acceleration in flow direction is expressed as br sin a and the centripetal acceleration in the flow direction is described as x2r cos a, i.e. x2y. For the different tubes of natural circulation, the angular acceleration in the flow direction is bli, where li is the distance between the tube and rolling axis.
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5.4. Modification terms Considering the experimental data, the average flow velocity in the rolling motion case is lower than that in the non-rolling case (Figs. 4 and 5). However, the computational results according to the above model show the average flow velocity of the natural circulation loop in the rolling case is the same as that in the non-rolling case. This may be because CR = 1.0 is assumed. This assumption results in the frictional resistance coefficient being calculated by the normal method without consideration of rolling motion. In other words, in the present model, the frictional resistance coefficient in the rolling case is presumed to be the same as that in the non-rolling case. The rolling motion and coolant flow fluctuation influence not only the heat transfer coefficient but also the frictional resistance coefficient. Therefore, the frictional resistance coefficient should be modified. For a flow fluctuation, there are three important parameters to describe the flow velocity, which are the maximum (m+), minimum Þ. With different values for CR, (m) and average flow velocities ðm . Using experimental data, one can obtain different values for m the values of CR for different rolling cases are obtained and shown in Table 2. An empirical correlation (Eq. (28)) is then obtained with an error less than ±4%, as shown in Fig. 13. Eq. (28) shows the frictional resistance coefficient for rolling motion is larger than that in the non-rolling case. The theoretically calculated RNCFV with the modification agrees well with the experimental data, as shown in Fig. 14 and Table 3.
C R ¼ 0:8942 þ 5:17265ðhm =t20 Þ ¼ 0:8942 þ 0:131bmax
Fig. 12. Force analysis for rolling case.
By integrating the added relative acceleration over each differential control volume, one can obtain the additional pressure drop caused by angular acceleration.
X
bli lj qi ¼ b
X
li lj qi
ð22Þ
The additional pressure drop caused by centripetal acceleration is expressed as
XZ
x2 zi qi dz þ
XZ
x2 yi qi dy ¼ x2
X Z
zi qi dz þ
XZ
yi qi dy
ð23Þ The Coriolis acceleration is perpendicular to the flow direction and thus cannot cause any additional pressure drop. 5.3. Energy conservation equation The energy conservation equation under rolling motion is written as
Pc ¼ P P in Pout
ð24Þ
where Pc is the heat capacity of the heated tube, P is the input power, Pin is the power absorbed by the work fluid, and Pout is power loss with Pout = 0 except for the heated section. They are expressed as
dT w dPc ¼ qw cpw Aw dL dt dPin ¼ ain ðT w TÞpDin dL dPout ¼ ae ðT w T 1 ÞpDout dL
ð25Þ ð26Þ
Eq. (28) shows the maximum angular acceleration bmax strongly influences the flow friction resistance coefficient under rolling motion. This result is very similar to the influence of the maximum angular acceleration on the heat transfer. Therefore, the maximum angular acceleration is the most important parameter influencing the characteristics of natural circulation flow under rolling motion. In fact, the heat transfer is not increased by rolling motion directly, but is enhanced by the flow fluctuation. If the rolling motion cannot vary the flow velocity, it may not influence the heat transfer. Such conclusions were also found by Pendyala et al. [30,31]. 6. Discussion 6.1. Effect of rolling motion on flow velocity The calculated average flow velocities under different rolling motion conditions are the same as those in the non-rolling case and the average RNCFV is 1.0, if the influence factor of the rolling motion CR is unchanged. However, the calculated average RNCFV agrees well with the experimental RNCFV with a modified CR. The average flow velocity under rolling motion and average RNCFV decrease as a result of the flow frictional coefficient being modified. Thus the increase in flow friction is the main factor in decreasTable 2 Coefficient CR in different rolling case. hm (°)
t0 (s)
CR
10
12.5 10 7.5
1.27 1.44 1.87
15
12.5 10 7.5
1.36 1.66 2.31
20
12.5 10 7.5
1.5 1.86 2.7
ð27Þ
where ain is calculated using Eq. (13) and ae is calculated using methods in the literature [32]. Thus a set of equations has been developed based on the mass, momentum and energy conservation equations and the available constitutive equations, and the equations can be solved by the Runge–Kutta method.
ð28Þ
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2.8 2.6 2.4
CR-cal
2.2 2.0 1.8 1.6 1.4 1.2 1.0 1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
CR-exp Fig. 13. Comparison between experimental and calculated values for CR.
calculation without modification calculation with modification experimental data
1.5 1.4 1.3
v r /v
1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0
10
20
30
40
t /s
creases under rolling motion, because of the increase in the frictional pressure drop. The fluctuation amplitude increases with increasing rolling amplitude and frequency. This is because (a) the additional acceleration and additional pressure drops increase and (b) the degree of the incline increases with increasing rolling amplitude, the effect of horizontal parts of the natural circulation loop increases, and thus the variation of the driving head increases. With increasing rolling amplitude and frequency, the amplitude of fluctuation increases, and thus the crest point of flow fluctuation increases and the trough point of flow fluctuation decreases. However, the average flow velocity decreases, so the trough point decreases greatly and the crest point increases only a little. Hence, the increment in the wave crest points of the relative flow velocity is smaller than the decrease in amplitude of the trough points and the influence of rolling motion is greater on the trough point of the flow fluctuation than on the wave crest point, as shown in Figs. 6 and 7. 6.2. Effects of angular and centripetal accelerations From Eqs. (3) and (22), in one rolling period, the direction of angular acceleration changes with time and there is a consistent direction of flow around the loop at any given time because the rolling axis is along the middle of the loop. Thus the effects of angular acceleration on different parts accumulate. From Eqs. (2) and (23), the direction of centripetal acceleration does not change with time. Because the rolling axis is along the middle of the loop and the loop is symmetrical, the effects of centripetal acceleration on different parts mostly counteract one another. Therefore, angular acceleration is the primary influence of rolling motion on the natural circulation within our experimental system and such a conclusion can be proved by Eqs. (16) and (28). Furthermore, Eqs. (13) and (28) cannot be extended to calculate the heat transfer coefficient and flow frictional coefficient for other experimental systems. However, the equations indicate that natural circulation flow is influenced by the acceleration induced by the rolling motion, and that heat transfer is influenced by the additional flow. 6.3. Effect of inlet subcooling of the test section
Fig. 14. Comparison of calculated and experimental data.
ing the average RNCFV. CR increases with increasing rolling amplitude and frequency. Thus the average RNCFV decreases with increasing rolling amplitude and frequency. Furthermore, the increase in flow friction is the reason why the outlet pressure de-
The amplitude of flow fluctuation decreases with decreasing inlet subcooling of the test section as shown in Figs. 6 and 7. If the inlet subcooling decreases, then the density difference between hot and cold legs increases and the average density of the whole loop decreases, so the amplitude of the additional pressure drop decreases and the effect of the driving head increases. Therefore,
Table 3 Comparisons between the calculated and experimental results. Rolling amplitude/period
me =m0
mc =m0
mþe =m0
mþc =m0
me =m0
mc =m0
20°/12.5 s
0.897 0.891 0.937
0.882 0.889 0.897
1.315 1.306 1.298
1.393 1.370 1.352
0.345 0.426 0.456
0.371 0.409 0.443
20°/10 s
0.785 0.792 0.838 0.856
0.789 0.795 0.802 0.827
1.280 1.262 1.262 1.223
1.428 1.397 1.371 1.275
0.214 0.262 0.319 0.396
0.149 0.193 0.233 0.269
10°/10 s
0.884 0.908 0.918
0.892 0.909 0.926
1.203 1.242 1.154
1.201 1.186 1.176
0.555 0.680 0.665
0.583 0.633 0.676
10°/7.5 s
0.743 0.809 0.830
0.786 0.798 0.810
1.055 1.113 1.112
1.199 1.168 1.144
0.388 0.495 0.576
0.375 0.430 0.477
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under the same experimental conditions, the amplitude of fluctuation decreases with increasing inlet temperature.
[9]
6.4. Effect of system pressure [10]
The liquid coolant is distilled water in the experiments. The water density almost does not vary in the range of experimental pressure. Thus the system pressure does not influence the coolant fluctuation.
[11] [12] [13]
7. Conclusions [14]
This paper experimentally and theoretically studies the singlephase natural circulation flow and heat transfer under a rolling motion condition. The rolling motion makes the fluid flow fluctuate periodically. The period of fluctuation is the same as the rolling period, and the fluctuation is close to being a sine wave. The fluctuation amplitude increases with increases in the rolling amplitude, rolling frequency and inlet subcooling. The primary influencing factors of the rolling motion are additional accelerations, among which angular acceleration is paramount. The increase in the friction resistance coefficient caused by flow fluctuation is the main reason why the average RNCFV decreases under rolling motion. The rolling motion is beneficial to heat transfer. The heat transfer coefficient in the rolling case is higher than that in the non-rolling case. Eq. (13) is suitable for calculating the heat transfer coefficient for natural circulation under rolling motion in the parameter ranges of the present study. A mathematical model was developed to simulate the natural circulation flow and heat transfer under rolling motion with an acceptable relative error.
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
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