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CFD analysis of thermal–hydraulic behavior in SCWR typical flow channels
Nuclear Engineering and Design 238 (2008) 3348–3359
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CFD analysis of thermal–hydraulic behavior in SCWR typical flow channels H.Y. Gu ∗ , X. Cheng, Y.H. Yang School of Nuclear Science and Engineering, Shanghai Jiao Tong University, 800 Dong Chuang Road, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 27 September 2007 Received in revised form 4 June 2008 Accepted 17 June 2008
a b s t r a c t Investigations on thermal–hydraulic behavior in SCWR fuel assembly have obtained a significant attention in the international SCWR community. However, there is still a lack of understanding and ability to predict the heat transfer behavior of supercritical water. In this paper, CFD analysis is carried out to study the flow and heat transfer behavior of supercritical water in sub-channels of both square and triangular rod bundles. Effect of various parameters, e.g. thermal boundary conditions and pitch-to-diameter ratio on the thermal–hydraulic behavior is investigated. Two boundary conditions, i.e., constant heat flux at the outer surface of cladding and constant heat density in the fuel pin are applied. The results show that the structure of the secondary flow mainly depends on the rod bundle configuration as well as the pitch-to-diameter ratio, whereas, the amplitude of the secondary flow is affected by the thermal boundary conditions, as well. The secondary flow is much stronger in a square lattice than that in a triangular lattice. The turbulence behavior is similar in both square and triangular lattices. The dependence of the amplitude of the turbulent velocity fluctuation across the gap on Reynolds number becomes prominent in both lattices as the pitch-to-diameter ratio increases. The effect of thermal boundary conditions on turbulent velocity fluctuation is negligibly small. For both lattices with small pitch-to-diameter ratios (P/D < 1.3), the mixing coefficient is about 0.022. Both secondary flow and turbulent mixing show unusual behavior in the vicinity of the pseudo-critical point. Further investigation is needed. A strong circumferential non-uniformity of wall temperature and heat transfer is observed in tight lattices at constant heat flux boundary conditions, especially in square lattices. In the case with constant heat density of fuel pin, the circumferential conductive heat transfer significantly reduces the non-uniformity of circumferential distribution of wall temperature and heat transfer, which is favorable for the design of SCWR fuel assemblies. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Supercritical water-cooled reactor (SCWR) is characterized as low flow rate, high enthalpy rise, and single-phase water-cooling, by which high thermal efficiency is achieved. Design of SCWR is based on the proven LWR technology and the experiences of fossil power plants with supercritical steam conditions. The role of SCWR has been emphasized on economic electricity generation for nearor mid-term nuclear market. The SCWR has been regarded as an innovative reactor and selected as one of candidates of Generation IV reactor systems. Research activities are ongoing worldwide to develop advanced nuclear power plants with SCWR (Oka and Koshizuka, 2000; Yoo et al., 2005; Kamei et al., 2006; Schulenberg and Starflinger, 2007; Oka et al., 2007). One of the main features of supercritical water is the strong variation of its thermal–physical properties in the vicinity
∗ Corresponding author. Tel.: +86 21 3420 4917. E-mail address: [email protected] (H.Y. Gu). 0029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2008.06.010
of the pseudo-critical line. This large variation of thermal–physical properties results in an unusual flow and heat transfer behavior. So the reliable knowledge of the thermal–hydraulic behavior at reactor relevant conditions is very important for the design of the SCWR core. Studies of thermal–hydraulic behavior of supercritical fluids have been performed since the 1950s. The existing experimental and theoretical studies on heat transfer at supercritical pressure conditions were reviewed and published by several authors (Jackson and Hall, 1979; Polyakov, 1991; Cheng and Schulenberg, 2001; Pioro et al., 2004; Pioro and Duffey, 2005). Exhaustive literature search, carried out by Pioro and Duffey (2005), showed that the majority of experimental data were obtained in vertical tubes. An experimental investigation on heat transfer in larger bundles at supercritical pressures was performed by Silin et al. (1993). It was found that the experimental heat transfer data were satisfactorily described by correlations obtained for water flow in tubes at supercritical pressures for the normal heat transfer regime, but no heat transfer deterioration (HTD) in rod bundles were observed within the same test parameter range for which heat transfer deterioration occurred in tubes. It should be noticed that the experimental
H.Y. Gu et al. / Nuclear Engineering and Design 238 (2008) 3348–3359
Nomenclature A c D f h K l m Nu p q Re t T u U v v V w W ¯ W
sub-channel flow area (m2 ) coefficient in Eq. (14) diameter (m) friction factor heat transfer coefficient (W/m2 ◦ C) dimensionless secondary flow energy in Eq. (3) length (m) mass flux (kg/m2 s) Nusselt number rod pitch (m) heat flux (W/m2 ) Reynolds number time (s) temperature (◦ C) or time interval (s) shear stress velocity (m/s) velocity along X axis (m/s) velocity fluctuation across the gap (m/s) deviation of v from its average amplitude (m/s) velocity along Y axis (m/s) radial width of cladding (m) velocity along Z axis (m/s) mean velocity along Z axis (m/s)
Greek symbols ˇ turbulent mixing coefficient ı gas gap (m) ε dimensionless circumferential temperature distribution thermal conductivity (W/m ◦ C) circumferential angle (◦ ) standard deviation in the Gaussian profile stress (N/m2 ) Subscripts B bulk c circumferential eq equivalent f fluid fuel fuel pellet gas gas gap gap gap region in fuel rod in inner surface of cladding N normalized r radial sec secondary flow symm symmetrical region in cladding w wall
investigations devoted to heat transfer in bundles cooled with water at supercritical pressures are very limited; more work is greatly needed to provide reliable information for design purposes. Due to the limitation in the experimental measurement techniques, numerical investigations using Computational Fluid Dynamics (CFD) codes has achieved a significant attention with the purpose to provide a better understanding of the heat transfer mechanism. The main difficulties in the numerical analysis are related to the turbulence modeling at supercritical pressures. In the earlier works, turbulence modelling was carried out by the simple eddy diffusivity approach (Shiralkar and Griffith, 1969). Qualitatively, a good agreement between the numerical prediction and the experimental data has been obtained. In spite of quantita-
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tive deviation between the numerical results and the experimental data, CFD is proven as a suitable approach for the analysis of the thermal–hydraulic behavior of supercritical fluids. With the development of the computer capability in recent years, advanced turbulence models have been applied in more and more numerical studies. Due to a sharp variation of properties near the heated wall, either a fine numerical mesh structure or a suitable wall treatment is necessary. During the last few years, in the frame of the development of supercritical water-cooled reactors (SCWR), extensive efforts have been made to assess the applicability of existing CFD codes to the heat transfer simulation at supercritical pressure conditions. The numerical analysis using more than 10 first order closure turbulence models was compared to experimental data for upward flows in circular tube by Kim et al. (2004). The results showed that RNG k–ε model with enhanced near-wall treatment gives the most outstanding prediction. This was also confirmed by Roelof and Komen (2005). Yang et al. (2007) numerically investigated the heat transfer in upward flows of supercritical water in circular tubes using 13 turbulent models with different wall treatment in STAR-CD3.24. Their results showed both two-layer model (Hassid and Poreh, 1978) and standard k–ε high Re model with the standard wall function give acceptable prediction when compared with experimental data and correlations. In the flow channels other than a circular tube, e.g. the typical flow channels of a SCWR fuel assembly, anisotropic behavior of turbulence and secondary flow occur (Kjellström, 1971; Trupp and Azad, 1975; Vonka, 1988; Rehme, 1992). Thus, turbulence models, which are capable of the simulation of anisotropic behavior of turbulence, are highly required for the analysis of heat transfer in sub-channels. Recently, Cheng et al. (2007) has studied the effect of mesh structures and turbulence models on heat transfer of supercritical water using CFX5.6. Based on the assessment in circular tubes and due to the requirement on the capability of simulating the anisotropic behavior of turbulence, the SSG Reynolds stress model is recommended for the application to sub-channel geometries. Although a large number of numerical studies on the heat transfer of supercritical fluids have been carried out by various authors, previous works are mostly limited to simple flow channel geometries, e.g. circular tubes. It is well known that conclusions achieved in circular tubes cannot be directly extrapolated to non-circular flow channel, due to different flow patterns. Recently, CFD analysis has been performed to investigate the thermal–hydraulic behavior of supercritical water in sub-channels of both square-array and triangular-array rod bundles using CFX5.6 by Cheng et al. (2007). The secondary flow was reproduced and the amplitude of secondary flow agreed qualitatively well with the experimental data of Carajilescov and Todreas (1976). It was also found that the average amplitude of the turbulent velocity fluctuation across the gap is weakly dependent on Reynolds number and pitch-to-diameter ratio in their calculation parameter ranges. Remarkably, Cheng et al. (2007) found the phenomenon of strong non-uniformity of the circumferential distribution of the cladding surface temperature and heat transfer. The non-uniformity of heat transfer is more significant in a tighter lattice than in a wider one, especially in square lattice. In their numerical analysis, Yang et al. (2007) confirmed the results of Cheng et al. (2007) and proposed design measures to reduce the non-uniformity. It should be noted that in the numerical analysis mentioned above, boundary condition of constant heat flux was applied and the conductive heat transfer in cladding was neglected. To provide basic understanding of thermal–hydraulic behavior in supercritical water at SCWR conditions, in the present study a systematical analysis is carried out using the CFD code CFX5.6. Sub-channels of both square and triangular lattices are selected. The effect of various parameters, e.g. geometric configuration and
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thermal boundary conditions on the thermal–hydraulic behavior, including flow pattern, turbulence mixing and heat transfer, is investigated. 2. Computational procedures In the present study, regular sub-channels of bare rod bundles are considered, both in square and in triangular arrangement, as indicated in Fig. 1. Due to the symmetric feature, only 1/4 of a regular sub-channel of the square lattice and 1/3 of a regular subchannel of the triangular lattice are taken as computational domain for the CFD analysis. Fuel rod parameters are selected based on the design parameters presented by Yoo et al. (2006) and Hofmeister et al. (2007). The values of the fuel rod diameter and fuel pellet diameter used are 8.0 mm and 7.04 mm, respectively. In calculation, the Reynolds number at the inlet of the sub-channel is set to 51,000. Selection of other operating parameters is based on the parameters
Table 1 Parameters used for the CFX simulation Parameters
Range
Rod bundle arrangement Cladding material Outer cladding diameter [mm] Inner cladding diameter [mm] Gas gap [mm] Fuel pellet diameter [mm] Pitch-to-diameter ratio Rod length [m] Pressure [MPa] Mass flux [kg/m2 s] Wall heat flux [kW/m2 ] Rod power [kW/m] Volume heat density [kW/m3 ] Fluid bulk temperature [◦ C] Reynolds number Turbulence models
Square, triangular Stainless steel 8.0 7.14 0.05 7.04 1.1–1.6 4.0 25.0 350–2000 600 15.07 387,400 280–510 51,000–500,000 SSG
widely used in SCWRs (Cheng et al., 2007). The ranges of various parameters used in this study are summarized in Table 1. In addition to boundary conditions of constant heat flux at the outer surface of cladding, boundary conditions of constant heat density in fuel pin is also considered. In this case, thermal conduction in fuel rod is coupled with convective heat transfer to fluid. It is assumed that heat is uniformly produced in fuel pellet and transferred to flowing supercritical water through gas gap and cladding by heat conduction. The thermal conductivity of UO2 fuel pellet with a 95% theoretical density is calculated as below (Yu et al., 2001): fuel =
3824 + 4.788 × 10−11 (T + 273.15)3 T + 402.55
(1)
where T is the local temperature of fuel pellet. The equivalent heat transfer coefficient in gas gap is set to heq = 5678 W/(m2 ◦ C), as widely used in PWR design (Yu et al., 2001). Therefore, the thermal conductivity in gas gap can be obtained as below: gas = heq × ı = 0.28 W (m ◦ C)
(2) 20 W/(m ◦ C)
Thermal conductivity of cladding is set to (Weisman, 1977). The thermal physical properties of water used in this paper are taken from Wagner and Kruse (1998). In all simulation cases, the total heat transferred to water is the same. The second order closure turbulence model SSG is applied, which, among all four ε-type models in CFX5.6, is the only one turbulence model capable of simulating the anisotropic behavior of turbulence. 3. Results and discussions 3.1. Flow pattern
Fig. 1. Sub-channels configuration. (a) Square lattice. (b) Triangular lattice.
Fig. 2 shows the projection of velocity vector on the X–Y plane in square and triangular lattices at various pitch-to-diameter ratios. The bulk fluid temperature on the elevation is 384 ◦ C, very close to the pseudo-critical value. The boundary condition of constant heat flux at outer surface of cladding is adopted. As seen that in all rod bundles secondary flow appears. One secondary flow cell is found in each one-eighth sub-channel of the tight square lattice with P/D = 1.2, while two secondary flow cells are found in the wide square lattice with P/D = 1.4. Considering that the mean axial velocity is 2.3 m/s in the square lattice with P/D = 1.2 and 1.27 m/s with P/D = 1.4, the maximum secondary flow velocity in the tight square lattice is about 0.36% of the mean axial velocity, lower than that in the wide lattice, which is about 0.63% of the mean axial velocity. In the triangular lattices of both P/D = 1.2 and P/D = 1.4 only one secondary flow cell is found in each one-sixth sub-channel. The
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Fig. 2. Flow patterns in square and triangular bundle lattices (Tb = 384 ◦ C). (a) P/D = 1.2, square lattice. (b) P/D = 1.4, square lattice. (c) P/D = 1.2, triangular lattice. (d) P/D = 1.4, square lattice.
maximum secondary flow velocity in the tight triangular lattice of P/D = 1.2 is about 0.15% of the mean axial velocity, lower than that in the wide lattice of P/D = 1.4, which is about 0.22% of the mean axial velocity. Obviously, the secondary flow in a square lattice is stronger than that in a triangular lattice. These results qualitatively agree well with the findings of the experimental works of Carajilescov and Todreas (1976) and Trupp and Azad (1975) and numerical simulation of Cheng and Tak (2006). In addition to the boundary conditions of constant heat flux and constant volume heat density in fuel pellet, CFD analyses with constant wall surface temperature Tw = 500 ◦ C is carried out, to figure out the effect of thermal boundary conditions on the secondary flow characteristics. It has to be pointed out that in the case of constant wall surface temperature, the total heat transferred to water is not the same as that under other two conditions. For all
cases shown in Fig. 3, the bulk fluid temperature on the elevation is 384 ◦ C. The secondary flow patterns are very similar at various boundary conditions. There exist two secondary flow cells in each one-eighth sub-channel. The secondary flow velocity is 0.63% of the mean axial velocity at constant wall surface hear flux, 0.51% at constant heat density in fuel pellet, and 0.57% at constant wall surface temperature. To present the amplitude of secondary flow as a function of bulk temperature clearly, the parameter, dimensionless secondary flow energy, is introduced and defined as below: K=
(1/A) (1/A)
2 dA mUsec m(U 2 + V 2 ) dA A = A 2 2 A
mW dA
A
mW dA
where U, V, W is the velocity along X, Y, and Z axis, respectively.
(3)
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Fig. 3. Effect of boundary conditions on flow patterns in square lattice (Tb = 384 ◦ C, P/D = 1.4). (a) Constant volume heat density in fuel pellet. (b) Constant heat flux at cladding outer surface. (c) Constant temperature at cladding outer surface (Tw = 500 ◦ C).
Fig. 4 shows the dimensionless secondary flow energy versus the bulk temperature in the square lattice of P/D = 1.1 and 1.4, respectively. In the tight square lattice with P/D = 1.1 as shown in Fig. 4a, the secondary flow energy decreases with increasing bulk temperature at a bulk temperature lower than the pseudo-critical value. The minimum secondary flow energy is achieved when the fluid temperature approaches the pseudo-critical value. The secondary flow energy increases as the bulk temperature increases when the bulk temperature exceeds the pseudo-critical value. In addition, lower secondary flow energy is obtained under the boundary condition of constant heat flux. For example, when the fluid bulk temperature is 384 ◦ C, the secondary flow energy at constant heat flux is 49% of that at constant volume heat density in fuel pellet. In contrary to the result for the tight square lattice, the secondary flow energy
increases at first and then decreases with the increase of bulk temperature for the wide square lattice with the maximum value occurring at pseudo-critical temperature. In the pseudo-critical region, the secondary flow energy at boundary conditions of constant volume heat density in fuel pellet is lower than that under constant wall surface heat flux. The effect of pitch-to-diameter ratio on the secondary flow energy in a triangular lattice is similar to that in a square lattice, as shown in Fig. 5. However, in a wide lattice, the secondary flow energy is much lower in the triangular lattice than that in the square lattice. For example, at bulk temperature of 384 ◦ C, the secondary flow energy in the triangular lattice is only 20% of that in the square lattice at the same pitch-to-diameter ratio (P/D = 1.4) under constant wall surface heat flux.
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Fig. 4. Secondary flow energy in square lattice at various P/D ratios. (a) P/D = 1.1. (b) P/D = 1.4.
4. Turbulent mixing From the definition of the Reynolds stress, which can be directly provided by the CFX calculation, one can obtain the relationship between the Reynolds stress vv vv =
1 T
v2 dt
(4)
T
and the average amplitude of the velocity fluctuation across the gap |¯v| |¯v| =
1 T
|v| dt
(5)
T
vv =
1 T
Fig. 5. Secondary flow energy in triangular lattice at various P/D ratios. (a) P/D = 1.1. (b) P/D = 1.4.
approximation for the velocity fluctuation (Lin, 1961). Therefore, in this study, a Gaussian distribution of the probability distribution for the velocity fluctuation is assumed, i.e. 1 2 2 e−v /2 f (v) = √ 2
v2 dt = T
1 T
¯ 2 + v ) dt = |v| ¯ 2+ (|v| 2
T
1 T
¯ 2 (6) v dt ≥ |v| 2
T
In Eq. (6), T stands for the time interval considered, t for time, v for velocity fluctuation and v for the deviation of the velocity fluctuation amplitude from the average value of the fluctuation amplitude, as defined below: ¯ = |v| − v = |v| − |v|
1 T
T
|v| dt
(7)
0
The statistical feature of the velocity fluctuation in turbulent flows depends on flow conditions. However, according to some experimental information available, the Gaussian distribution of the probability distribution was found, in many cases, to be a fairly good
(8)
Using the probability density function, it yields
∞
v2 · f (v) dv =
vv = 0
and
|¯v| = 0
It yields
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∞
2 2
v · f (v) dv = √ 2
From Eqs. (9) and (10) it yields √ vv ¯ = √ |v|
(9)
(10)
(11)
Eq. (11) is used in the present study to derive the average amplitude of the velocity fluctuation across the gap by knowing the Reynolds stress, which is obtained from the CFD analysis. Fig. 6 shows the average amplitude of the velocity fluctuation ¯ across the gap normalized by the shear velocity, i.e. c = |v|/u in the square lattice of P/D = 1.2 at two different thermal boundary conditions. The shear velocity u is calculated by
u =
w =
f ¯ W 8
(12)
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Fig. 6. Normalized velocity fluctuation across the gap at various conditions.
¯ axial mean flow velocity Here w is wall shear stress, density, W and f friction factor, which can be calculated using the conventional Blasius correlation f =
0.3164 Re0.25
(13)
It has to be pointed out that Eq. (13) is valid for Re < 105 . However, its application is extrapolated to the entire parameter range considered in the present paper. The result presented in Fig. 6 implies that the effect of thermal boundary conditions on normalized velocity fluctuation across the gap is negligibly small. The average amplitude of the velocity fluctuation across the gap has its maximum close to the wall. It decreases with the distance to the wall. Fig. 7 provides the normalized velocity fluctuation at the gap as function of Reynolds number at various pitch-to-diameter ratios. It can be found that Reynolds number has an apparent effect on normalized velocity fluctuation at the gap in a square lattice, especially in a wide square lattice. When the bulk temperature is close to pseudo-critical value (384 ◦ C) with the corresponding Reynolds number 123,000, the normalized velocity fluctuation shows its maximum value in the square lattice of P/D = 1.2, while it has the minimum value in the wide square lattice of P/D = 1.4. The effect of Reynolds number on the normalized velocity fluctuation at the gap is negligibly small in the tight triangular lattice (P/D = 1.2), but significant in the wide triangular lattice (P/D = 1.4) as shown in Fig. 8. In the wide triangular lattice (P/D = 1.4), the minimum normalized velocity fluctuation occurs at a bulk temperature close to the pseudo-critical value as well. According to the definition of the so-called turbulent mixing coefficient widely used in sub-channel analysis codes (Stewart, 1977), the mixing coefficient can be derived as ˇ=
|¯v| c · u = = c · 0.20 · Re−0.125 ¯ ¯ W W
(14)
Eq. (14) has the similar form as that proposed by Rogers and Todreas (1968). Fig. 9 shows the mixing coefficients across the gap versus bulk fluid temperature at various pitch-to-diameter ratios. Generally, CFD code gives similar behavior of mixing coefficients for both square and triangular arrangements. In general, at a pitchto-diameter ratio smaller than 1.3, the dependence of the mixing coefficient on the pitch-to-diameter ratio and bulk temperature is small, except that an abrupt fluctuation with small amplitude occurs in the vicinity of the pseudo-critical region. On the average, the mixing coefficient is about 0.022, larger than the value calculated using the Roger’s equation (0.008). In a wide lattice (P/D > 1.3), the mixing coefficient dramatically decreases when bulk temperature approaches the pseudo-critical value. This unusual behaviour needs further investigations. The mixing coefficient depends on
Fig. 7. Effect of Reynolds number on normalized velocity fluctuation at the gap in square lattice. (a) P/D = 1.1. (b) P/D = 1.4.
sub-channel geometry as well as flow parameters as discussed by Cheng and Muller (1998). Further investigation with a wider range of parameters is needed. 4.1. Temperature distribution Fig. 10 presents the temperature distribution at the outer surface of cladding in square lattice of P/D = 1.1 under two thermal boundary conditions. A strong circumferential non-uniformity of temperature distribution on cladding surface is observed under the boundary condition of constant wall heat flux, as shown in Fig. 10a. At the channel outlet cross-section, the wall temperature in the gap ( = 0◦ ) is about 150 ◦ C higher than that in the symmetrical region ( = 45◦ ). To achieve a more uniform distribution of wall temperatures, Yang et al. (2007) proposed some modification to the flow channel geometry. This makes, however, the mechanical design of the fuel assembly more complex. The circumferential wall temperature distribution is much more uniform at boundary conditions of constant heat density of the fuel pin, as shown in Fig. 10b. In this case, the wall temperature difference at the channel outlet decreases to about 30 ◦ C. At the same time, the maximum wall temperature in the gap is 88 ◦ C lower than that obtained under a constant wall surface heat flux. Therefore, the conductive heat transfer in the rod bundle can significantly improve the uniformity of the wall temperature distribution. Fig. 11 shows the local heat flux at the inner and outer surface of cladding versus the bulk temperature in a square lattice of P/D = 1.1 with constant volume heat density in fuel pellet. The local heat flux at the inner and outer surface is normalized by the corresponding
H.Y. Gu et al. / Nuclear Engineering and Design 238 (2008) 3348–3359
Fig. 8. Effect of Reynolds number on normalized velocity fluctuation at the gap in triangular lattice. (a) P/D = 1.1. (b) P/D = 1.4.
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Fig. 9. Effect of pitch-to-diameter ratio on turbulent mixing coefficient. (a) Square lattice. (b) Triangular lattice.
Here, the equivalent radial and circumferential heat transfer coefficients are given as: average surface heat flux. In this figure, results on the rod surface along three lines are presented, i.e. in the gap ( = 0◦ ), on the center line ( = 22.5◦ ) and on the symmetrical line ( = 45◦ ). The heat flux distribution at the inner surface of cladding is well uniform. On the average, the heat flux on the symmetrical line is about 10% higher than that in the gap. The heat flux along the symmetrical line ( = 45◦ ) at the cladding outer surface is much higher than that in the gap ( = 0◦ ). The heat flux on the symmetrical line is 54% higher than that in the gap at a fluid temperature far lower than the pseudo-critical temperature, and 88% at a fluid temperature far higher than the pseudo-critical temperature. This result indicates that in the case with a constant heat flux in fuel pellet, the redistribution of the circumferential heat flux is mainly achieved through the circumferential conductive heat transfer in cladding. Fig. 12 schematically shows the heat transfer process through the cladding in a square lattice of P/D = 1.1. Heat is radically transferred from the inner surface to the outer, and then further transferred to the fluid by convection. At the same time, the heat is also circumferentially transferred from the gap region ( = 0◦ ) to the symmetrical region ( = 45◦ ) driven by the circumferential temperature gradient in cladding. In order to qualitatively examine the effect of circumferential conductive heat transfer on the wall heat flux redistribution in cladding, the heat transferred radically and circumferentially is characterized as following: qr ∝ hr l(Tin − Tf )
(15a)
qc ∝ hc w(Tgap − Tsymm )
(15b)
hr =
1 (1/h) + (w/)
(16a)
hc =
= 10.3 kW/m2 ◦ C l
(16b)
where, is the thermal conductivity of cladding, h is convective heat transfer coefficient, l, w, are circumferential length and radial width of cladding, Tin , Tgap , Tsymm , Tf are average temperature of the inner surface of cladding, the gap region in fuel rod, the symmetrical region in cladding and fluid in channel at a cross-section, respectively. Fig. 13 gives the equivalent radial heat transfer coefficients along three lines ( = 0◦ , = 22.5◦ and = 45◦ ) and the equivalent circumferential heat transfer coefficients in cladding in the square lattice of P/D = 1.1. The amplitude of the circumferential heat transfer coefficient is comparable to that of the radial heat transfer coefficients, when the fluid temperature is far lower or higher than the pseudo-critical value. The non-uniformity of the circumferential wall temperature distribution is strong, as indicated under boundary conditions of constant heat flux. Driven by the large circumferential temperature gradient, the circumferential conductive heat transfer is significant and results in a higher heat flux in the symmetrical region than that in the gap region. However, in the region close to the pseudo-critical point, the radial heat transfer coefficient is considerably higher than the circumferential heat transfer coefficient. In this region, the heat transferred by circumferential conductive heat transfer is less important compared to that by the radial heat transfer through cladding. Therefore, the
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Fig. 11. Heat flux distribution at inner and outer surface of cladding in square lattice (P/D = 1.1). (a) Heat flux at the cladding inner surface. (b) Heat flux at the cladding outer surface.
or the onset of heat transfer deterioration (HTD) in the gap region is observed when the bulk fluid temperature approaches or exceeds the pseudo-critical value under the boundary condition of constant heat flux. At a high bulk fluid temperature, circumferential conductive heat transfer in cladding becomes significant, which leads to
Fig. 10. Temperature distribution at outer surface of cladding in square lattice at P/D = 1.1 (X:Y:Z = 1:1:660). (a) Constant wall heat flux. (b) Constant volume heat density. (c)
circumferential heat flux distribution in cladding becomes well uniform in the pseudo-critical region, as shown in Fig. 11b. Fig. 14 shows the local wall temperature at outer surface of cladding versus the bulk fluid temperature in square lattices with different pitch-to-diameter ratios under two thermal boundary conditions. In the figures, results on the rod surface along two lines are presented, i.e. in the gap ( = 0◦ ), on the symmetrical line ( = 45◦ ). The wall temperature in a tight lattice is lower than that in a wide lattice at the same bulk fluid temperature. This is mainly due to a smaller hydraulic diameter and a higher flow velocity in tighter lattice. A significant increase in the wall surface temperature
Fig. 12. Schematic diagram of heat transfer through cladding.
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Fig. 13. Circumferential and conduction heat transfer coefficient.
a reduction in the wall temperature of about 80 ◦ C at P/D = 1.1 and about 25 ◦ C at P/D = 1.4 in the gap ( = 0◦ ), and an increase of about 20 ◦ C on the symmetrical line ( = 45◦ ). Fig. 15 presents the azimuthal distribution of the temperature at the cladding outer surface in both lattices with different pitchto-diameter ratios. The bulk fluid temperature on this elevation is 468 ◦ C. The temperature in Fig. 15 is normalized as below: Normalized temperature TN =
Tw − TB T w,max − TB
(17)
Where Tw is the wall surface temperature, TB is the fluid bulk temperature and Tw,max is the maximum circumferential wall temperature. The curves show a strong effect of the pitch-to-diameter
Fig. 15. Circumferential distribution of normalized temperature at the cladding outer surface. (a) Square lattice. (b) Triangular lattice.
ratio on the circumferential temperature distribution. The nonuniformity of the circumferential temperature distribution in a tight lattice is much stronger than that in a wide lattice, especially in the square arrangement. The non-uniformity of circumferential temperature distribution is characterized as following: ε=
(TN )=0◦ − (TN )=45◦ (TN )=0◦
(18)
In the case of constant wall heat flux, the non-uniformity decreases from about 72% at P/D = 1.1 down to 40% at P/D = 1.2 in square lattice, and from 50% to 15% in triangular lattice. When the circumferential conductive heat transfer in cladding is taken into consideration, the circumferential temperature distribution becomes more uniform. The non-uniformity decreases to 30% at P/D = 1.1 and to 10% at P/D = 1.2 in the square lattice. In the triangular lattice, the nonuniformity decreases to 17% and 3%, respectively. Therefore, in rod bundles with a pitch-to-diameter ratio larger than 1.2, the nonuniformity of circumferential temperature distribution in the outer surface of the cladding is less prominent when the circumferential conductive heat transfer is taken into account. Fig. 16 presents the effect of the circumferential conductive heat transfer in cladding on the maximum surface temperature at the channel outlet cross-section with various pitch-to-diameter ratios. The normalized maximum temperature in Fig. 16 is defined as below: Normalized maximum temperature TN =
Fig. 14. Wall temperature at various circumferential angles in square lattice. (a) Angle = 0◦ . (b) Angle = 45◦ .
(T w,max − TB )with conduction (Tw,max − TB )without conduction
(19)
In a tight lattice, the non-uniformity of the circumferential temperature distribution is strong. The circumferential conductive heat transfer leads to a significant reduction in the maximum temperature in the gap region and, subsequently, the non-uniformity. At a
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Fig. 16. Effect of pitch-to-diameter ratio on maximum temperature at outer surface of cladding.
same pitch-to-diameter ratio, the reduction in the maximum wall temperature in a square lattice is more than that in a triangular lattice, e.g. at P/D = 1.2 the normalized temperature is 0.54 in the square lattice and 0.74 in the triangular lattice. With the increase in the pitch-to-diameter ratio, the nonuniformity of circumferential temperature distribution is greatly weakened, and the effect of conductive heat transfer in the cladding on the maximum wall temperature decreases. When a pitch-todiameter ratio is larger than 1.6 in a square lattice and 1.3 in a triangular lattice, the normalized temperatures are higher than 0.97. The effect of conductive heat transfer can be neglected.
Fig. 18. Circumferential distribution of Nusselt number at various P/D ratios in both lattices. (a) Square lattice. (b) Triangular lattice.
4.2. Heat transfer
Fig. 17. Nusselt number at various circumferential angles. (a) Square lattice. (b) Triangular lattice.
Fig. 17 shows the local Nusselt number versus the bulk fluid temperature in both square and triangular lattices with a pitch-todiameter ratio of 1.1. In this figure, results on the rod surface along three lines are presented, i.e. in the gap ( = 0◦ ), on the symmetrical line ( = 45◦ ) for square and 30◦ for triangular lattice), and on the center line ( = 22.5◦ ) for square and 15◦ for triangular lattice). A circumferentially non-uniform distribution of the Nusselt number is clearly identified in both lattices. Generally, heat transfer in the gap ( = 0◦ ) is less efficient. Heat transfer is enhanced by increasing the angle from the gap to the symmetric line. This non-uniformity is much stronger in a square lattice than that in a triangular lattice. At low bulk temperatures, i.e. lower than the pseudo-critical value, the equivalent radial heat transfer coefficient is much higher than the equivalent circumferential heat transfer coefficient in cladding as presented in Fig. 13, which leads to relatively small circumferential heat conduction in cladding. Consequently, at low bulk temperatures, the effect of thermal boundary conditions on local Nusselt number is small. However, in the case of high bulk temperatures, a strong reduction in the equivalent radial heat transfer coefficient occurs due to the HTD phenomena, which results in a significant increase in the circumferential heat conduction from the gap region to symmetrical region in cladding. A much less non-uniformity in the circumferential distribution of Nusselt number is obtained. At high fluid temperature, the ratio of Nusselt number on the symmetric line to that in the gap decreases from about 3.8 and 1.9 down to 2.65 and 1.67 for the square and triangular lattice, respectively, when the circumferential thermal conduction is considered. Fig. 18 compares the calculated Nusselt number in both lattice arrangements with different pitch-to-diameter ratios. Results are shown on an elevation where the bulk fluid temperature is 472 ◦ C and the local Reynolds number 177,000. A much stronger non-
H.Y. Gu et al. / Nuclear Engineering and Design 238 (2008) 3348–3359
uniformity in the circumferential distribution of Nusselt number is obtained in the tighter square lattice. The circumferential conductive heat transfer can significantly improve the uniformity in circumferential distribution of Nusselt number. The ratio of maximum and minimum circumferential Nusselt number decreases from 3.6 and 1.9 down to 2.6 and 1.6 in the square and triangular lattice of P/D = 1.1, respectively. 5. Summary CFD analysis has been carried out to achieve basic knowledge of supercritical water flows in typical flow channels of SCWR fuel assemblies. The thermal–hydraulic behavior in the sub-channels of both square lattice and triangular lattice is analyzed. The effect of the rod bundle configuration as well as the pitch-to-diameter ratio is investigated. Two thermal boundary conditions, i.e. constant heat flux at outer surface of cladding and constant volume heat density in fuel pellet are carried out. Some main results achieved can be summarized as follows: (1) The structure of secondary flow cell mainly depends on geometric configuration of sub-channels as well as pitch-to-diameter ratio. The secondary flow energy in a square lattice is much larger than that in a triangular lattice at the same pitch-todiameter ratio, except in a very tight lattice (P/D = 1.1). The secondary flow energy shows its maximum value at the pseudocritical temperature in a tight lattice (P/D < 1.3), while it has the minimum value at the pseudo-critical temperature in a wide lattice (P/D ≥ 1.3). (2) The turbulent velocity fluctuation across the gap is an important parameter determining the inter-channel exchange. The results show that the velocity fluctuation is similar in both square and triangular lattices. It has the maximum value close to the wall surface and decrease with the distance to the wall. The velocity fluctuation is slightly dependent on Reynolds number in tight lattices (P/D < 1.2), whereas significantly in wide lattices (P/D > 1.2). The effect of thermal boundary conditions on velocity fluctuation is negligibly small. In both lattice arrangements with small pitch-to-diameter ratios (P/D < 1.3), it is found that the turbulent mixing coefficient is about 0.022. An unusual behavior of turbulent mixing coefficient appears at a bulk temperature close to the pseudo-critical point. Further investigation is needed. (3) A strong circumferential non-uniformity of wall temperature and heat transfer is observed in tight lattices, especially in square lattices, when a constant heat flux boundary condition is applied. When the conductive heat transfer in cladding is taken into consideration, the circumferential conductive heat transfer in cladding significantly reduces the non-uniformity of the circumferential temperature and heat transfer distribution, and consequently, the maximum wall temperature decreases. The result is favorable for the design of SCWR fuel assemblies. Finally, it has to be pointed out that the conclusions achieved in this paper are only valid for the parameter ranges considered. Further analysis with a wider range of parameters is needed and underway. Acknowledgements The authors are grateful to National Basic Research Program of China (No. 2007CB209804) for providing the financial support for this study.
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Fuel assembly design study for a reactor with supercritical water. Nucl. Eng. Des. 237, 1513–1521. Jackson, J.D., Hall, W.B., 1979. Forced convection heat transfer to fluids at supercritical pressure. Turbulent Forced Convection in Channels and Bundles, vol. 2. Hemisphere, New York, pp. 563–611. Kamei, K., Yamaji, A., Ishiwatari, Y., OKa, Y., Liu, J., 2006. Fuel and core design of super light water reactor with low leakage fuel loading pattern. J. Nucl. Sci. Technol. 43, 129–219. Kim, S.H., Kim, Y.I., Bae, Y.Y., Cho, B.H., 2004. Numerical simulation of the vertical upward flow of water in a heated tube at supercritical pressure. In: Proc. ICAPP. Kjellström, B., 1971. Studies of turbulent flow parallel to a rod bundle of triangular array. Report AE-RV-196, AB Atomenergi, Sweden. Lin, C.C., 1961. Statistical theory of turbulence. Princeton University Press, Princeton, New Jersey. Oka, Y., Ishiwatari, Y., Koshizuka, S., 2007. Research and development of super LWR and super fast reactor. In: Proceeding of SCWR-2007, Shanghai, China, March 12–15, pp. 9–18. Oka, Y., Koshizuka, S., 2000. Design concept of once-through cycle supercritical pressure light water cooled reactors. In: Proceeding of SCR, Tokyo, Japan, November 6–8, pp. 1–22. Pioro, I.L., Duffey, R.B., 2005. Experimental heat transfer in supercritical water flowing inside channels (survey). Nucl. Eng. Des. 235, 2407–2430. Pioro, I.L., Khartabil, H.F., Duffey, R.B., 2004. Heat transfer to supercritical fluids flowing in channels-empirical correlations (survey). Nucl. Eng. Des. 230, 69–91. Polyakov, A.F., 1991. Heat transfer under supercritical pressure. Adv. Heat Transfer 21, 1–51. Rehme, K., 1992. The prediction of turbulent Prandtl and Schmidt numbers. Int. J. Heat Mass Transfer 18, 1055–1069. Roelof, F., Komen (Eds.), 2005. CFD analysis of heat transfer to supercritical water flowing vertically upward in a tube, Jahrestagung Kerntechnik, May 10–12, 2005. Nürnberg, Germany. Rogers, J.T., Todreas, N.E., 1968. Coolant inter-channel mixing in reactor fuel rod bundles single-phase coolant, in section heat transfer in rod bundle. In: The Winter Annual Meeting of ASME, New York, December, pp. 1–56. Schulenberg, T., Starflinger, J., 2007. European research project on high performance light water reactors. In: Proceeding of SCWR-2007, Shanghai, China, March 12–15, pp. 1–8. Shiralkar, B.S., Griffith, P., 1969. Deterioration in heat transfer to fluids at supercritical pressure and high heat fluxes. J. Heat Transfer (February), 27–36. Silin, V.A., Voznesensky, V.A., Afrov, A.M., 1993. The light water integral reactor with natural circulation of the coolant at supercritical pressure B-500 SKDI. Nucl. Eng. Des. 144, 327–336. Stewart, C.W., 1977. COBRA-IV: The models and the methods. Battelle Pacific Northwest Laboratories, Report BNWL-2214. Trupp, A.C., Azad, R.S., 1975. The structure of turbulent flow in triangular array rod bundles. Nucl. Eng. Des. 32, 47–84. Vonka, V., 1988. Measurement of secondary flow vortices in a rod bundle. Nucl. Eng. Des. 106, 191–207. Wagner, W., Kruse, A., 1998. The industrial standard IAPWS-IF97 for the thermodynamic properties and supplementary equations for other properties. In: Properties of Water and Steam. Springer-Verlag, Heidelberg. Weisman, J., 1977. Elements of Nuclear Reactor Design. Elsevier Scientific Publishing Company, Amsterdam. Yang, J., Oka, Y., Ishiwatari, Y., Liu, J., Yoo, J., 2007. Numerical investigation of heat transfer in upward flows of supercritical water in circular tubes and tight fuel rod bundles. Nucl. Eng. Des. 237, 420–430. Yoo, J., Ishiwatari, Y., Oka, Y., Liu, J., 2005. Composite core design of high power density supercritical water cooled fast reactor. In: Proceeding of GLOBAL2005, Tsukuba, Japan, October 9–13, 2005 (Paper No. 246). Yoo, J., Ishiwatari, Y., Oka, Y., Liu, J., 2006. Conceptual design of compact supercritical water-cooled fast reactor with thermal hydraulic coupling. Ann. Nucl. 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CFD analysis of thermal–hydraulic behavior in SCWR typical flow channels H.Y. Gu ∗ , X. Cheng, Y.H. Yang School of Nuclear Science and Engineering, Shanghai Jiao Tong University, 800 Dong Chuang Road, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 27 September 2007 Received in revised form 4 June 2008 Accepted 17 June 2008
a b s t r a c t Investigations on thermal–hydraulic behavior in SCWR fuel assembly have obtained a significant attention in the international SCWR community. However, there is still a lack of understanding and ability to predict the heat transfer behavior of supercritical water. In this paper, CFD analysis is carried out to study the flow and heat transfer behavior of supercritical water in sub-channels of both square and triangular rod bundles. Effect of various parameters, e.g. thermal boundary conditions and pitch-to-diameter ratio on the thermal–hydraulic behavior is investigated. Two boundary conditions, i.e., constant heat flux at the outer surface of cladding and constant heat density in the fuel pin are applied. The results show that the structure of the secondary flow mainly depends on the rod bundle configuration as well as the pitch-to-diameter ratio, whereas, the amplitude of the secondary flow is affected by the thermal boundary conditions, as well. The secondary flow is much stronger in a square lattice than that in a triangular lattice. The turbulence behavior is similar in both square and triangular lattices. The dependence of the amplitude of the turbulent velocity fluctuation across the gap on Reynolds number becomes prominent in both lattices as the pitch-to-diameter ratio increases. The effect of thermal boundary conditions on turbulent velocity fluctuation is negligibly small. For both lattices with small pitch-to-diameter ratios (P/D < 1.3), the mixing coefficient is about 0.022. Both secondary flow and turbulent mixing show unusual behavior in the vicinity of the pseudo-critical point. Further investigation is needed. A strong circumferential non-uniformity of wall temperature and heat transfer is observed in tight lattices at constant heat flux boundary conditions, especially in square lattices. In the case with constant heat density of fuel pin, the circumferential conductive heat transfer significantly reduces the non-uniformity of circumferential distribution of wall temperature and heat transfer, which is favorable for the design of SCWR fuel assemblies. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Supercritical water-cooled reactor (SCWR) is characterized as low flow rate, high enthalpy rise, and single-phase water-cooling, by which high thermal efficiency is achieved. Design of SCWR is based on the proven LWR technology and the experiences of fossil power plants with supercritical steam conditions. The role of SCWR has been emphasized on economic electricity generation for nearor mid-term nuclear market. The SCWR has been regarded as an innovative reactor and selected as one of candidates of Generation IV reactor systems. Research activities are ongoing worldwide to develop advanced nuclear power plants with SCWR (Oka and Koshizuka, 2000; Yoo et al., 2005; Kamei et al., 2006; Schulenberg and Starflinger, 2007; Oka et al., 2007). One of the main features of supercritical water is the strong variation of its thermal–physical properties in the vicinity
∗ Corresponding author. Tel.: +86 21 3420 4917. E-mail address: [email protected] (H.Y. Gu). 0029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2008.06.010
of the pseudo-critical line. This large variation of thermal–physical properties results in an unusual flow and heat transfer behavior. So the reliable knowledge of the thermal–hydraulic behavior at reactor relevant conditions is very important for the design of the SCWR core. Studies of thermal–hydraulic behavior of supercritical fluids have been performed since the 1950s. The existing experimental and theoretical studies on heat transfer at supercritical pressure conditions were reviewed and published by several authors (Jackson and Hall, 1979; Polyakov, 1991; Cheng and Schulenberg, 2001; Pioro et al., 2004; Pioro and Duffey, 2005). Exhaustive literature search, carried out by Pioro and Duffey (2005), showed that the majority of experimental data were obtained in vertical tubes. An experimental investigation on heat transfer in larger bundles at supercritical pressures was performed by Silin et al. (1993). It was found that the experimental heat transfer data were satisfactorily described by correlations obtained for water flow in tubes at supercritical pressures for the normal heat transfer regime, but no heat transfer deterioration (HTD) in rod bundles were observed within the same test parameter range for which heat transfer deterioration occurred in tubes. It should be noticed that the experimental
H.Y. Gu et al. / Nuclear Engineering and Design 238 (2008) 3348–3359
Nomenclature A c D f h K l m Nu p q Re t T u U v v V w W ¯ W
sub-channel flow area (m2 ) coefficient in Eq. (14) diameter (m) friction factor heat transfer coefficient (W/m2 ◦ C) dimensionless secondary flow energy in Eq. (3) length (m) mass flux (kg/m2 s) Nusselt number rod pitch (m) heat flux (W/m2 ) Reynolds number time (s) temperature (◦ C) or time interval (s) shear stress velocity (m/s) velocity along X axis (m/s) velocity fluctuation across the gap (m/s) deviation of v from its average amplitude (m/s) velocity along Y axis (m/s) radial width of cladding (m) velocity along Z axis (m/s) mean velocity along Z axis (m/s)
Greek symbols ˇ turbulent mixing coefficient ı gas gap (m) ε dimensionless circumferential temperature distribution thermal conductivity (W/m ◦ C) circumferential angle (◦ ) standard deviation in the Gaussian profile stress (N/m2 ) Subscripts B bulk c circumferential eq equivalent f fluid fuel fuel pellet gas gas gap gap gap region in fuel rod in inner surface of cladding N normalized r radial sec secondary flow symm symmetrical region in cladding w wall
investigations devoted to heat transfer in bundles cooled with water at supercritical pressures are very limited; more work is greatly needed to provide reliable information for design purposes. Due to the limitation in the experimental measurement techniques, numerical investigations using Computational Fluid Dynamics (CFD) codes has achieved a significant attention with the purpose to provide a better understanding of the heat transfer mechanism. The main difficulties in the numerical analysis are related to the turbulence modeling at supercritical pressures. In the earlier works, turbulence modelling was carried out by the simple eddy diffusivity approach (Shiralkar and Griffith, 1969). Qualitatively, a good agreement between the numerical prediction and the experimental data has been obtained. In spite of quantita-
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tive deviation between the numerical results and the experimental data, CFD is proven as a suitable approach for the analysis of the thermal–hydraulic behavior of supercritical fluids. With the development of the computer capability in recent years, advanced turbulence models have been applied in more and more numerical studies. Due to a sharp variation of properties near the heated wall, either a fine numerical mesh structure or a suitable wall treatment is necessary. During the last few years, in the frame of the development of supercritical water-cooled reactors (SCWR), extensive efforts have been made to assess the applicability of existing CFD codes to the heat transfer simulation at supercritical pressure conditions. The numerical analysis using more than 10 first order closure turbulence models was compared to experimental data for upward flows in circular tube by Kim et al. (2004). The results showed that RNG k–ε model with enhanced near-wall treatment gives the most outstanding prediction. This was also confirmed by Roelof and Komen (2005). Yang et al. (2007) numerically investigated the heat transfer in upward flows of supercritical water in circular tubes using 13 turbulent models with different wall treatment in STAR-CD3.24. Their results showed both two-layer model (Hassid and Poreh, 1978) and standard k–ε high Re model with the standard wall function give acceptable prediction when compared with experimental data and correlations. In the flow channels other than a circular tube, e.g. the typical flow channels of a SCWR fuel assembly, anisotropic behavior of turbulence and secondary flow occur (Kjellström, 1971; Trupp and Azad, 1975; Vonka, 1988; Rehme, 1992). Thus, turbulence models, which are capable of the simulation of anisotropic behavior of turbulence, are highly required for the analysis of heat transfer in sub-channels. Recently, Cheng et al. (2007) has studied the effect of mesh structures and turbulence models on heat transfer of supercritical water using CFX5.6. Based on the assessment in circular tubes and due to the requirement on the capability of simulating the anisotropic behavior of turbulence, the SSG Reynolds stress model is recommended for the application to sub-channel geometries. Although a large number of numerical studies on the heat transfer of supercritical fluids have been carried out by various authors, previous works are mostly limited to simple flow channel geometries, e.g. circular tubes. It is well known that conclusions achieved in circular tubes cannot be directly extrapolated to non-circular flow channel, due to different flow patterns. Recently, CFD analysis has been performed to investigate the thermal–hydraulic behavior of supercritical water in sub-channels of both square-array and triangular-array rod bundles using CFX5.6 by Cheng et al. (2007). The secondary flow was reproduced and the amplitude of secondary flow agreed qualitatively well with the experimental data of Carajilescov and Todreas (1976). It was also found that the average amplitude of the turbulent velocity fluctuation across the gap is weakly dependent on Reynolds number and pitch-to-diameter ratio in their calculation parameter ranges. Remarkably, Cheng et al. (2007) found the phenomenon of strong non-uniformity of the circumferential distribution of the cladding surface temperature and heat transfer. The non-uniformity of heat transfer is more significant in a tighter lattice than in a wider one, especially in square lattice. In their numerical analysis, Yang et al. (2007) confirmed the results of Cheng et al. (2007) and proposed design measures to reduce the non-uniformity. It should be noted that in the numerical analysis mentioned above, boundary condition of constant heat flux was applied and the conductive heat transfer in cladding was neglected. To provide basic understanding of thermal–hydraulic behavior in supercritical water at SCWR conditions, in the present study a systematical analysis is carried out using the CFD code CFX5.6. Sub-channels of both square and triangular lattices are selected. The effect of various parameters, e.g. geometric configuration and
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thermal boundary conditions on the thermal–hydraulic behavior, including flow pattern, turbulence mixing and heat transfer, is investigated. 2. Computational procedures In the present study, regular sub-channels of bare rod bundles are considered, both in square and in triangular arrangement, as indicated in Fig. 1. Due to the symmetric feature, only 1/4 of a regular sub-channel of the square lattice and 1/3 of a regular subchannel of the triangular lattice are taken as computational domain for the CFD analysis. Fuel rod parameters are selected based on the design parameters presented by Yoo et al. (2006) and Hofmeister et al. (2007). The values of the fuel rod diameter and fuel pellet diameter used are 8.0 mm and 7.04 mm, respectively. In calculation, the Reynolds number at the inlet of the sub-channel is set to 51,000. Selection of other operating parameters is based on the parameters
Table 1 Parameters used for the CFX simulation Parameters
Range
Rod bundle arrangement Cladding material Outer cladding diameter [mm] Inner cladding diameter [mm] Gas gap [mm] Fuel pellet diameter [mm] Pitch-to-diameter ratio Rod length [m] Pressure [MPa] Mass flux [kg/m2 s] Wall heat flux [kW/m2 ] Rod power [kW/m] Volume heat density [kW/m3 ] Fluid bulk temperature [◦ C] Reynolds number Turbulence models
Square, triangular Stainless steel 8.0 7.14 0.05 7.04 1.1–1.6 4.0 25.0 350–2000 600 15.07 387,400 280–510 51,000–500,000 SSG
widely used in SCWRs (Cheng et al., 2007). The ranges of various parameters used in this study are summarized in Table 1. In addition to boundary conditions of constant heat flux at the outer surface of cladding, boundary conditions of constant heat density in fuel pin is also considered. In this case, thermal conduction in fuel rod is coupled with convective heat transfer to fluid. It is assumed that heat is uniformly produced in fuel pellet and transferred to flowing supercritical water through gas gap and cladding by heat conduction. The thermal conductivity of UO2 fuel pellet with a 95% theoretical density is calculated as below (Yu et al., 2001): fuel =
3824 + 4.788 × 10−11 (T + 273.15)3 T + 402.55
(1)
where T is the local temperature of fuel pellet. The equivalent heat transfer coefficient in gas gap is set to heq = 5678 W/(m2 ◦ C), as widely used in PWR design (Yu et al., 2001). Therefore, the thermal conductivity in gas gap can be obtained as below: gas = heq × ı = 0.28 W (m ◦ C)
(2) 20 W/(m ◦ C)
Thermal conductivity of cladding is set to (Weisman, 1977). The thermal physical properties of water used in this paper are taken from Wagner and Kruse (1998). In all simulation cases, the total heat transferred to water is the same. The second order closure turbulence model SSG is applied, which, among all four ε-type models in CFX5.6, is the only one turbulence model capable of simulating the anisotropic behavior of turbulence. 3. Results and discussions 3.1. Flow pattern
Fig. 1. Sub-channels configuration. (a) Square lattice. (b) Triangular lattice.
Fig. 2 shows the projection of velocity vector on the X–Y plane in square and triangular lattices at various pitch-to-diameter ratios. The bulk fluid temperature on the elevation is 384 ◦ C, very close to the pseudo-critical value. The boundary condition of constant heat flux at outer surface of cladding is adopted. As seen that in all rod bundles secondary flow appears. One secondary flow cell is found in each one-eighth sub-channel of the tight square lattice with P/D = 1.2, while two secondary flow cells are found in the wide square lattice with P/D = 1.4. Considering that the mean axial velocity is 2.3 m/s in the square lattice with P/D = 1.2 and 1.27 m/s with P/D = 1.4, the maximum secondary flow velocity in the tight square lattice is about 0.36% of the mean axial velocity, lower than that in the wide lattice, which is about 0.63% of the mean axial velocity. In the triangular lattices of both P/D = 1.2 and P/D = 1.4 only one secondary flow cell is found in each one-sixth sub-channel. The
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Fig. 2. Flow patterns in square and triangular bundle lattices (Tb = 384 ◦ C). (a) P/D = 1.2, square lattice. (b) P/D = 1.4, square lattice. (c) P/D = 1.2, triangular lattice. (d) P/D = 1.4, square lattice.
maximum secondary flow velocity in the tight triangular lattice of P/D = 1.2 is about 0.15% of the mean axial velocity, lower than that in the wide lattice of P/D = 1.4, which is about 0.22% of the mean axial velocity. Obviously, the secondary flow in a square lattice is stronger than that in a triangular lattice. These results qualitatively agree well with the findings of the experimental works of Carajilescov and Todreas (1976) and Trupp and Azad (1975) and numerical simulation of Cheng and Tak (2006). In addition to the boundary conditions of constant heat flux and constant volume heat density in fuel pellet, CFD analyses with constant wall surface temperature Tw = 500 ◦ C is carried out, to figure out the effect of thermal boundary conditions on the secondary flow characteristics. It has to be pointed out that in the case of constant wall surface temperature, the total heat transferred to water is not the same as that under other two conditions. For all
cases shown in Fig. 3, the bulk fluid temperature on the elevation is 384 ◦ C. The secondary flow patterns are very similar at various boundary conditions. There exist two secondary flow cells in each one-eighth sub-channel. The secondary flow velocity is 0.63% of the mean axial velocity at constant wall surface hear flux, 0.51% at constant heat density in fuel pellet, and 0.57% at constant wall surface temperature. To present the amplitude of secondary flow as a function of bulk temperature clearly, the parameter, dimensionless secondary flow energy, is introduced and defined as below: K=
(1/A) (1/A)
2 dA mUsec m(U 2 + V 2 ) dA A = A 2 2 A
mW dA
A
mW dA
where U, V, W is the velocity along X, Y, and Z axis, respectively.
(3)
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Fig. 3. Effect of boundary conditions on flow patterns in square lattice (Tb = 384 ◦ C, P/D = 1.4). (a) Constant volume heat density in fuel pellet. (b) Constant heat flux at cladding outer surface. (c) Constant temperature at cladding outer surface (Tw = 500 ◦ C).
Fig. 4 shows the dimensionless secondary flow energy versus the bulk temperature in the square lattice of P/D = 1.1 and 1.4, respectively. In the tight square lattice with P/D = 1.1 as shown in Fig. 4a, the secondary flow energy decreases with increasing bulk temperature at a bulk temperature lower than the pseudo-critical value. The minimum secondary flow energy is achieved when the fluid temperature approaches the pseudo-critical value. The secondary flow energy increases as the bulk temperature increases when the bulk temperature exceeds the pseudo-critical value. In addition, lower secondary flow energy is obtained under the boundary condition of constant heat flux. For example, when the fluid bulk temperature is 384 ◦ C, the secondary flow energy at constant heat flux is 49% of that at constant volume heat density in fuel pellet. In contrary to the result for the tight square lattice, the secondary flow energy
increases at first and then decreases with the increase of bulk temperature for the wide square lattice with the maximum value occurring at pseudo-critical temperature. In the pseudo-critical region, the secondary flow energy at boundary conditions of constant volume heat density in fuel pellet is lower than that under constant wall surface heat flux. The effect of pitch-to-diameter ratio on the secondary flow energy in a triangular lattice is similar to that in a square lattice, as shown in Fig. 5. However, in a wide lattice, the secondary flow energy is much lower in the triangular lattice than that in the square lattice. For example, at bulk temperature of 384 ◦ C, the secondary flow energy in the triangular lattice is only 20% of that in the square lattice at the same pitch-to-diameter ratio (P/D = 1.4) under constant wall surface heat flux.
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Fig. 4. Secondary flow energy in square lattice at various P/D ratios. (a) P/D = 1.1. (b) P/D = 1.4.
4. Turbulent mixing From the definition of the Reynolds stress, which can be directly provided by the CFX calculation, one can obtain the relationship between the Reynolds stress vv vv =
1 T
v2 dt
(4)
T
and the average amplitude of the velocity fluctuation across the gap |¯v| |¯v| =
1 T
|v| dt
(5)
T
vv =
1 T
Fig. 5. Secondary flow energy in triangular lattice at various P/D ratios. (a) P/D = 1.1. (b) P/D = 1.4.
approximation for the velocity fluctuation (Lin, 1961). Therefore, in this study, a Gaussian distribution of the probability distribution for the velocity fluctuation is assumed, i.e. 1 2 2 e−v /2 f (v) = √ 2
v2 dt = T
1 T
¯ 2 + v ) dt = |v| ¯ 2+ (|v| 2
T
1 T
¯ 2 (6) v dt ≥ |v| 2
T
In Eq. (6), T stands for the time interval considered, t for time, v for velocity fluctuation and v for the deviation of the velocity fluctuation amplitude from the average value of the fluctuation amplitude, as defined below: ¯ = |v| − v = |v| − |v|
1 T
T
|v| dt
(7)
0
The statistical feature of the velocity fluctuation in turbulent flows depends on flow conditions. However, according to some experimental information available, the Gaussian distribution of the probability distribution was found, in many cases, to be a fairly good
(8)
Using the probability density function, it yields
∞
v2 · f (v) dv =
vv = 0
and
|¯v| = 0
It yields
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∞
2 2
v · f (v) dv = √ 2
From Eqs. (9) and (10) it yields √ vv ¯ = √ |v|
(9)
(10)
(11)
Eq. (11) is used in the present study to derive the average amplitude of the velocity fluctuation across the gap by knowing the Reynolds stress, which is obtained from the CFD analysis. Fig. 6 shows the average amplitude of the velocity fluctuation ¯ across the gap normalized by the shear velocity, i.e. c = |v|/u in the square lattice of P/D = 1.2 at two different thermal boundary conditions. The shear velocity u is calculated by
u =
w =
f ¯ W 8
(12)
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Fig. 6. Normalized velocity fluctuation across the gap at various conditions.
¯ axial mean flow velocity Here w is wall shear stress, density, W and f friction factor, which can be calculated using the conventional Blasius correlation f =
0.3164 Re0.25
(13)
It has to be pointed out that Eq. (13) is valid for Re < 105 . However, its application is extrapolated to the entire parameter range considered in the present paper. The result presented in Fig. 6 implies that the effect of thermal boundary conditions on normalized velocity fluctuation across the gap is negligibly small. The average amplitude of the velocity fluctuation across the gap has its maximum close to the wall. It decreases with the distance to the wall. Fig. 7 provides the normalized velocity fluctuation at the gap as function of Reynolds number at various pitch-to-diameter ratios. It can be found that Reynolds number has an apparent effect on normalized velocity fluctuation at the gap in a square lattice, especially in a wide square lattice. When the bulk temperature is close to pseudo-critical value (384 ◦ C) with the corresponding Reynolds number 123,000, the normalized velocity fluctuation shows its maximum value in the square lattice of P/D = 1.2, while it has the minimum value in the wide square lattice of P/D = 1.4. The effect of Reynolds number on the normalized velocity fluctuation at the gap is negligibly small in the tight triangular lattice (P/D = 1.2), but significant in the wide triangular lattice (P/D = 1.4) as shown in Fig. 8. In the wide triangular lattice (P/D = 1.4), the minimum normalized velocity fluctuation occurs at a bulk temperature close to the pseudo-critical value as well. According to the definition of the so-called turbulent mixing coefficient widely used in sub-channel analysis codes (Stewart, 1977), the mixing coefficient can be derived as ˇ=
|¯v| c · u = = c · 0.20 · Re−0.125 ¯ ¯ W W
(14)
Eq. (14) has the similar form as that proposed by Rogers and Todreas (1968). Fig. 9 shows the mixing coefficients across the gap versus bulk fluid temperature at various pitch-to-diameter ratios. Generally, CFD code gives similar behavior of mixing coefficients for both square and triangular arrangements. In general, at a pitchto-diameter ratio smaller than 1.3, the dependence of the mixing coefficient on the pitch-to-diameter ratio and bulk temperature is small, except that an abrupt fluctuation with small amplitude occurs in the vicinity of the pseudo-critical region. On the average, the mixing coefficient is about 0.022, larger than the value calculated using the Roger’s equation (0.008). In a wide lattice (P/D > 1.3), the mixing coefficient dramatically decreases when bulk temperature approaches the pseudo-critical value. This unusual behaviour needs further investigations. The mixing coefficient depends on
Fig. 7. Effect of Reynolds number on normalized velocity fluctuation at the gap in square lattice. (a) P/D = 1.1. (b) P/D = 1.4.
sub-channel geometry as well as flow parameters as discussed by Cheng and Muller (1998). Further investigation with a wider range of parameters is needed. 4.1. Temperature distribution Fig. 10 presents the temperature distribution at the outer surface of cladding in square lattice of P/D = 1.1 under two thermal boundary conditions. A strong circumferential non-uniformity of temperature distribution on cladding surface is observed under the boundary condition of constant wall heat flux, as shown in Fig. 10a. At the channel outlet cross-section, the wall temperature in the gap ( = 0◦ ) is about 150 ◦ C higher than that in the symmetrical region ( = 45◦ ). To achieve a more uniform distribution of wall temperatures, Yang et al. (2007) proposed some modification to the flow channel geometry. This makes, however, the mechanical design of the fuel assembly more complex. The circumferential wall temperature distribution is much more uniform at boundary conditions of constant heat density of the fuel pin, as shown in Fig. 10b. In this case, the wall temperature difference at the channel outlet decreases to about 30 ◦ C. At the same time, the maximum wall temperature in the gap is 88 ◦ C lower than that obtained under a constant wall surface heat flux. Therefore, the conductive heat transfer in the rod bundle can significantly improve the uniformity of the wall temperature distribution. Fig. 11 shows the local heat flux at the inner and outer surface of cladding versus the bulk temperature in a square lattice of P/D = 1.1 with constant volume heat density in fuel pellet. The local heat flux at the inner and outer surface is normalized by the corresponding
H.Y. Gu et al. / Nuclear Engineering and Design 238 (2008) 3348–3359
Fig. 8. Effect of Reynolds number on normalized velocity fluctuation at the gap in triangular lattice. (a) P/D = 1.1. (b) P/D = 1.4.
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Fig. 9. Effect of pitch-to-diameter ratio on turbulent mixing coefficient. (a) Square lattice. (b) Triangular lattice.
Here, the equivalent radial and circumferential heat transfer coefficients are given as: average surface heat flux. In this figure, results on the rod surface along three lines are presented, i.e. in the gap ( = 0◦ ), on the center line ( = 22.5◦ ) and on the symmetrical line ( = 45◦ ). The heat flux distribution at the inner surface of cladding is well uniform. On the average, the heat flux on the symmetrical line is about 10% higher than that in the gap. The heat flux along the symmetrical line ( = 45◦ ) at the cladding outer surface is much higher than that in the gap ( = 0◦ ). The heat flux on the symmetrical line is 54% higher than that in the gap at a fluid temperature far lower than the pseudo-critical temperature, and 88% at a fluid temperature far higher than the pseudo-critical temperature. This result indicates that in the case with a constant heat flux in fuel pellet, the redistribution of the circumferential heat flux is mainly achieved through the circumferential conductive heat transfer in cladding. Fig. 12 schematically shows the heat transfer process through the cladding in a square lattice of P/D = 1.1. Heat is radically transferred from the inner surface to the outer, and then further transferred to the fluid by convection. At the same time, the heat is also circumferentially transferred from the gap region ( = 0◦ ) to the symmetrical region ( = 45◦ ) driven by the circumferential temperature gradient in cladding. In order to qualitatively examine the effect of circumferential conductive heat transfer on the wall heat flux redistribution in cladding, the heat transferred radically and circumferentially is characterized as following: qr ∝ hr l(Tin − Tf )
(15a)
qc ∝ hc w(Tgap − Tsymm )
(15b)
hr =
1 (1/h) + (w/)
(16a)
hc =
= 10.3 kW/m2 ◦ C l
(16b)
where, is the thermal conductivity of cladding, h is convective heat transfer coefficient, l, w, are circumferential length and radial width of cladding, Tin , Tgap , Tsymm , Tf are average temperature of the inner surface of cladding, the gap region in fuel rod, the symmetrical region in cladding and fluid in channel at a cross-section, respectively. Fig. 13 gives the equivalent radial heat transfer coefficients along three lines ( = 0◦ , = 22.5◦ and = 45◦ ) and the equivalent circumferential heat transfer coefficients in cladding in the square lattice of P/D = 1.1. The amplitude of the circumferential heat transfer coefficient is comparable to that of the radial heat transfer coefficients, when the fluid temperature is far lower or higher than the pseudo-critical value. The non-uniformity of the circumferential wall temperature distribution is strong, as indicated under boundary conditions of constant heat flux. Driven by the large circumferential temperature gradient, the circumferential conductive heat transfer is significant and results in a higher heat flux in the symmetrical region than that in the gap region. However, in the region close to the pseudo-critical point, the radial heat transfer coefficient is considerably higher than the circumferential heat transfer coefficient. In this region, the heat transferred by circumferential conductive heat transfer is less important compared to that by the radial heat transfer through cladding. Therefore, the
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Fig. 11. Heat flux distribution at inner and outer surface of cladding in square lattice (P/D = 1.1). (a) Heat flux at the cladding inner surface. (b) Heat flux at the cladding outer surface.
or the onset of heat transfer deterioration (HTD) in the gap region is observed when the bulk fluid temperature approaches or exceeds the pseudo-critical value under the boundary condition of constant heat flux. At a high bulk fluid temperature, circumferential conductive heat transfer in cladding becomes significant, which leads to
Fig. 10. Temperature distribution at outer surface of cladding in square lattice at P/D = 1.1 (X:Y:Z = 1:1:660). (a) Constant wall heat flux. (b) Constant volume heat density. (c)
circumferential heat flux distribution in cladding becomes well uniform in the pseudo-critical region, as shown in Fig. 11b. Fig. 14 shows the local wall temperature at outer surface of cladding versus the bulk fluid temperature in square lattices with different pitch-to-diameter ratios under two thermal boundary conditions. In the figures, results on the rod surface along two lines are presented, i.e. in the gap ( = 0◦ ), on the symmetrical line ( = 45◦ ). The wall temperature in a tight lattice is lower than that in a wide lattice at the same bulk fluid temperature. This is mainly due to a smaller hydraulic diameter and a higher flow velocity in tighter lattice. A significant increase in the wall surface temperature
Fig. 12. Schematic diagram of heat transfer through cladding.
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Fig. 13. Circumferential and conduction heat transfer coefficient.
a reduction in the wall temperature of about 80 ◦ C at P/D = 1.1 and about 25 ◦ C at P/D = 1.4 in the gap ( = 0◦ ), and an increase of about 20 ◦ C on the symmetrical line ( = 45◦ ). Fig. 15 presents the azimuthal distribution of the temperature at the cladding outer surface in both lattices with different pitchto-diameter ratios. The bulk fluid temperature on this elevation is 468 ◦ C. The temperature in Fig. 15 is normalized as below: Normalized temperature TN =
Tw − TB T w,max − TB
(17)
Where Tw is the wall surface temperature, TB is the fluid bulk temperature and Tw,max is the maximum circumferential wall temperature. The curves show a strong effect of the pitch-to-diameter
Fig. 15. Circumferential distribution of normalized temperature at the cladding outer surface. (a) Square lattice. (b) Triangular lattice.
ratio on the circumferential temperature distribution. The nonuniformity of the circumferential temperature distribution in a tight lattice is much stronger than that in a wide lattice, especially in the square arrangement. The non-uniformity of circumferential temperature distribution is characterized as following: ε=
(TN )=0◦ − (TN )=45◦ (TN )=0◦
(18)
In the case of constant wall heat flux, the non-uniformity decreases from about 72% at P/D = 1.1 down to 40% at P/D = 1.2 in square lattice, and from 50% to 15% in triangular lattice. When the circumferential conductive heat transfer in cladding is taken into consideration, the circumferential temperature distribution becomes more uniform. The non-uniformity decreases to 30% at P/D = 1.1 and to 10% at P/D = 1.2 in the square lattice. In the triangular lattice, the nonuniformity decreases to 17% and 3%, respectively. Therefore, in rod bundles with a pitch-to-diameter ratio larger than 1.2, the nonuniformity of circumferential temperature distribution in the outer surface of the cladding is less prominent when the circumferential conductive heat transfer is taken into account. Fig. 16 presents the effect of the circumferential conductive heat transfer in cladding on the maximum surface temperature at the channel outlet cross-section with various pitch-to-diameter ratios. The normalized maximum temperature in Fig. 16 is defined as below: Normalized maximum temperature TN =
Fig. 14. Wall temperature at various circumferential angles in square lattice. (a) Angle = 0◦ . (b) Angle = 45◦ .
(T w,max − TB )with conduction (Tw,max − TB )without conduction
(19)
In a tight lattice, the non-uniformity of the circumferential temperature distribution is strong. The circumferential conductive heat transfer leads to a significant reduction in the maximum temperature in the gap region and, subsequently, the non-uniformity. At a
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Fig. 16. Effect of pitch-to-diameter ratio on maximum temperature at outer surface of cladding.
same pitch-to-diameter ratio, the reduction in the maximum wall temperature in a square lattice is more than that in a triangular lattice, e.g. at P/D = 1.2 the normalized temperature is 0.54 in the square lattice and 0.74 in the triangular lattice. With the increase in the pitch-to-diameter ratio, the nonuniformity of circumferential temperature distribution is greatly weakened, and the effect of conductive heat transfer in the cladding on the maximum wall temperature decreases. When a pitch-todiameter ratio is larger than 1.6 in a square lattice and 1.3 in a triangular lattice, the normalized temperatures are higher than 0.97. The effect of conductive heat transfer can be neglected.
Fig. 18. Circumferential distribution of Nusselt number at various P/D ratios in both lattices. (a) Square lattice. (b) Triangular lattice.
4.2. Heat transfer
Fig. 17. Nusselt number at various circumferential angles. (a) Square lattice. (b) Triangular lattice.
Fig. 17 shows the local Nusselt number versus the bulk fluid temperature in both square and triangular lattices with a pitch-todiameter ratio of 1.1. In this figure, results on the rod surface along three lines are presented, i.e. in the gap ( = 0◦ ), on the symmetrical line ( = 45◦ ) for square and 30◦ for triangular lattice), and on the center line ( = 22.5◦ ) for square and 15◦ for triangular lattice). A circumferentially non-uniform distribution of the Nusselt number is clearly identified in both lattices. Generally, heat transfer in the gap ( = 0◦ ) is less efficient. Heat transfer is enhanced by increasing the angle from the gap to the symmetric line. This non-uniformity is much stronger in a square lattice than that in a triangular lattice. At low bulk temperatures, i.e. lower than the pseudo-critical value, the equivalent radial heat transfer coefficient is much higher than the equivalent circumferential heat transfer coefficient in cladding as presented in Fig. 13, which leads to relatively small circumferential heat conduction in cladding. Consequently, at low bulk temperatures, the effect of thermal boundary conditions on local Nusselt number is small. However, in the case of high bulk temperatures, a strong reduction in the equivalent radial heat transfer coefficient occurs due to the HTD phenomena, which results in a significant increase in the circumferential heat conduction from the gap region to symmetrical region in cladding. A much less non-uniformity in the circumferential distribution of Nusselt number is obtained. At high fluid temperature, the ratio of Nusselt number on the symmetric line to that in the gap decreases from about 3.8 and 1.9 down to 2.65 and 1.67 for the square and triangular lattice, respectively, when the circumferential thermal conduction is considered. Fig. 18 compares the calculated Nusselt number in both lattice arrangements with different pitch-to-diameter ratios. Results are shown on an elevation where the bulk fluid temperature is 472 ◦ C and the local Reynolds number 177,000. A much stronger non-
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uniformity in the circumferential distribution of Nusselt number is obtained in the tighter square lattice. The circumferential conductive heat transfer can significantly improve the uniformity in circumferential distribution of Nusselt number. The ratio of maximum and minimum circumferential Nusselt number decreases from 3.6 and 1.9 down to 2.6 and 1.6 in the square and triangular lattice of P/D = 1.1, respectively. 5. Summary CFD analysis has been carried out to achieve basic knowledge of supercritical water flows in typical flow channels of SCWR fuel assemblies. The thermal–hydraulic behavior in the sub-channels of both square lattice and triangular lattice is analyzed. The effect of the rod bundle configuration as well as the pitch-to-diameter ratio is investigated. Two thermal boundary conditions, i.e. constant heat flux at outer surface of cladding and constant volume heat density in fuel pellet are carried out. Some main results achieved can be summarized as follows: (1) The structure of secondary flow cell mainly depends on geometric configuration of sub-channels as well as pitch-to-diameter ratio. The secondary flow energy in a square lattice is much larger than that in a triangular lattice at the same pitch-todiameter ratio, except in a very tight lattice (P/D = 1.1). The secondary flow energy shows its maximum value at the pseudocritical temperature in a tight lattice (P/D < 1.3), while it has the minimum value at the pseudo-critical temperature in a wide lattice (P/D ≥ 1.3). (2) The turbulent velocity fluctuation across the gap is an important parameter determining the inter-channel exchange. The results show that the velocity fluctuation is similar in both square and triangular lattices. It has the maximum value close to the wall surface and decrease with the distance to the wall. The velocity fluctuation is slightly dependent on Reynolds number in tight lattices (P/D < 1.2), whereas significantly in wide lattices (P/D > 1.2). The effect of thermal boundary conditions on velocity fluctuation is negligibly small. In both lattice arrangements with small pitch-to-diameter ratios (P/D < 1.3), it is found that the turbulent mixing coefficient is about 0.022. An unusual behavior of turbulent mixing coefficient appears at a bulk temperature close to the pseudo-critical point. Further investigation is needed. (3) A strong circumferential non-uniformity of wall temperature and heat transfer is observed in tight lattices, especially in square lattices, when a constant heat flux boundary condition is applied. When the conductive heat transfer in cladding is taken into consideration, the circumferential conductive heat transfer in cladding significantly reduces the non-uniformity of the circumferential temperature and heat transfer distribution, and consequently, the maximum wall temperature decreases. The result is favorable for the design of SCWR fuel assemblies. Finally, it has to be pointed out that the conclusions achieved in this paper are only valid for the parameter ranges considered. Further analysis with a wider range of parameters is needed and underway. Acknowledgements The authors are grateful to National Basic Research Program of China (No. 2007CB209804) for providing the financial support for this study.
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