Heat transfer deterioration to supercritical water in circular tube and annular channel

Nuclear Engineering and Design 255 (2013) 97–104

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Heat transfer deterioration to supercritical water in circular tube and annular channel Lei Liu, Zejun Xiao ∗ , Xiao Yan, Xiaokang Zeng, Yanping Huang CNNC Key Laboratory on Nuclear Reactor Thermal Hydraulics Technology, Nuclear Power Institute of China, Southern 3rd Section, 2nd Ring Road, Chengdu, Sichuan, PR China

h i g h l i g h t s    

Numerical simulation was implemented on heat transfer deterioration in circular tube and annular channel. Influence of tube diameter on heat transfer deterioration was studied. Comparison of heat transfer deterioration in circular tube and annular channel was carried out. Influence of thermal equivalent diameter on heat transfer deterioration was investigated.

a r t i c l e

i n f o

Article history: Received 10 February 2012 Received in revised form 7 August 2012 Accepted 4 September 2012

a b s t r a c t In order to ensure the safety of operation in Supercritical Water-Cooled nuclear Reactor (SCWR), heat transfer deterioration (HTD) must be avoided. In this paper, HTD was numerically studied in circular tube and annular channel using eight low-Reynolds-number models to provide detailed information on flow and turbulence structure. All models considered are able to predict HTD phenomena, while SST model performs better than other models quantitatively. Numerical results indicate that acceleration effect is significantly important for HTD at high mass flux while buoyancy effect at low mass flux. The increase of tube diameter leads to the aggravation of HTD phenomena, which is more obvious at low mass flux. At high mass flux, HTD phenomenon in circular tube is severer than that in inner-wall-heated annular channel. However, with both walls heated in annular channel, HTD phenomenon is similar to that in circular tube. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Supercritical Water-Cooled nuclear Reactor (SCWR) is highpressure (∼25 MPa) and high-temperature (outlet temperature is above 500 ◦ C) reactor that will operate above the thermodynamic critical point of water (22.1 MPa, 374 ◦ C), and the coolant would pass through the pseudo-critical region before reaching the outlet (Jacopo, 2003). There is no liquid–vapor phase transition at supercritical pressures, so critical heat flux or dryout will not occur. However, the specific thermal physical properties of supercritical water could result in some special features such as heat transfer deterioration (HTD) (Jackson and Hall, 1979; Pioro and Duffey, 2005; Kao et al., 2010; Wen and Gu, 2011). The works devoted to heat transfer characteristics of supercritical water were started as early as the 1950s, and a number of investigations on HTD phenomena in simple channels have been performed so far.

∗ Corresponding author. Tel.: +86 028 85908901; fax: +86 028 85908889. E-mail address: fabulous [email protected] (Z. Xiao). 0029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nucengdes.2012.09.025

Shitsman (1963) carried out an experiment in vertical circular tube and first observed the phenomenon of heat transfer deterioration at supercritical pressures at low mass flux with relatively high heat flux. Krasyakova (1967) observed deteriorated heat transfer in a horizontal tube. However, the temperature profile for a horizontal tube at locations of deteriorated heat transfer differs from that for a vertical tube, being smoother for a horizontal tube compared to that of a vertical tube with a higher temperature increase on the upper part of the tube than on the lower part. Ornatskiy et al. (1970) investigated the appearance of deteriorated heat transfer in five parallel tubes. They found that the deteriorated heat transfer in the assembly at supercritical pressures depended on the heat-flux to mass-flux ratio and flow conditions. They also established the possibility of the simultaneous existence of several local zones of deteriorated heat transfer along the tubes. Glushchenko and Gandzyuk (1972) conducted experiments with an upward flow of water in annular channel with external and internal one-side heating. In general, the results of the investigation showed that variations in wall temperature of a heated tube and of an annulus, when the tubes and annuli are fairly long, were similar. However, in annuli with normal and deteriorated heat

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Nomenclature Abbreviations 2D two dimensional 3D three dimensional abid AB AKN Abe–Kondoh–Nagano CFD Computational Fluid Dynamics Chang–Hsieh–Chen CHC HTD heat transfer deterioration Lam–Bremhors LB LS Launder–Sharma Super Critical Water-cooled Reactor SCWR SST Shear Stress Transport YS Yang–Shih V2F k − ε − v2 − f General symbols A flowing area (m2 ) buoyancy parameter Bu Cf friction coefficient Cp specific heat capacity (kJ kg−1 K−1 ) c , c1 , c2 model constants tube diameter (m)/dumping function D Dh hydraulic equivalent diameter (m) dh axial enthalpy gradient (kJ kg−1 m−1 ) dz dp+ acceleration parameter a dpa pressure gradient caused by acceleration (Pa/m) dz dpf dz dpg dz dptot dz DT

E f , f1 , f2 G Gr g h htot k L p p Ph Pr q Re SE SM Tin Tpc U u u˜ u ⊗ u uh y+

pressure gradient caused by friction (Pa/m) pressure gradient caused by gravity (Pa/m) total pressure gradient (Pa/m) thermal equivalent diameter (m) dumping function dumping function mass flux (kg/m2 s) Grashof number gravity acceleration (m s−2 ) enthalpy (kJ/kg) total enthalpy (kJ/kg) turbulence kinetic energy (m2 s−2 ) tube length (m) operating pressure (MPa) effective pressure (MPa) heating circumference (m) Prandtl number heat flux (kW/m2 ) Reynolds number energy source term momentum source term inlet temperature (◦ C) pseudo critical temperature (◦ C) velocity vector (m/s) fluctuating velocity (m/s) mass weighted average velocity (m/s) Reynolds stress (m2 s−2 ) additional turbulence flux (kJ/kg) non-dimensional distance from the heated wall

Greek letters closure coefficient ˇ∗ ε rate of dissipation of k (m2 s−3 ) thermal conductivity (W m−1 K−1 )   kinematic viscosity (m2 /s)

 eff t  ¯ t  ω

dynamic viscosity (kg m−1 s−1 ) effective viscosity (kg m−1 s−1 ) Eddy viscosity (kg m−1 s−1) density (kg m−3 ) integrated density (kg m−3 ) Eddy diffusivity (kg m−1 s−1 ) shear stress (kg m−1 s−2 ) specific dissipation rate (s−1 )

Subscripts b bulk in i o out

transfer no decrease in temperature (past the zone of deteriorated heat transfer) was noticed in their experiments. Ornatskiy et al. (1972) investigated normal and deteriorated heat transfer in a vertical annulus. The deteriorated heat transfer zone was observed visually as a red hot spot, appearing in the upper section of the test tube. The hot spot elongated in the direction of the annulus inlet with increasing heat flux. Due to the limitation in the experimental measurement techniques, detailed information on experimental flow field and turbulence is rare for supercritical water. In recent years, with the development of technologies in computational methods, numerical investigations using Computational Fluid Dynamics (CFD) codes have made significant achievement with the purpose to obtain prediction of heat transfer deterioration in supercritical water. Koshizuka et al. (1995) performed a 2D numerical analysis for heat transfer of supercritical water in a 10 mm circular tube. The results agreed well with the test data of Yamagata et al. (1972). Based on the numerical results, an empirical correlation of heat transfer coefficient was obtained. Palko and Anglart (2008) used SST model to make simulations of HTD. The results showed that the influence of buoyancy forces on the heat transfer is significant at low mass flux. At high mass flux, acceleration and thermal physical properties of coolant are more important for the decrease of heat transfer. Gu et al. (2008) studied HTD phenomenon numerically using six low-Reynolds-number models including AKN, CHC, ABID, YS, V2F and SST. The results showed that all models considered are to some extent able to reproduce the effect of buoyancy on heat transfer. V2F and SST models perform better than other models in predicting the onset of deterioration. Kao et al. (2010) concluded the important roles of buoyancy effect and Prandtl number on HTD of supercritical water inside circular tube using CFD method. Series of simulations with various operational pressure and inlet temperature indicated that the increase of both inlet temperature and operational pressure were very effective to relax HTD. Bae et al. (2010) carried out an experiment of heat transfer to CO2 , which flows upward and downward in a circular tube. Effect of tube diameter on supercritical heat transfer was studied with inner diameter of 4.4 mm, 6.0 mm, and 9.0 mm. The results show that when the mass flux is kept constant, the heat transfer rate increases with the increasing diameter. Bae (2011) carried out a series of experiments with CO2 flowing upward and downward in a circular tube with an inner diameter of 4.57 mm and an annular channel created between a tube with an inner diameter of 10 mm and a heater rod with an outer diameter of 8 mm. Comparison between a tube and annular channel with the same thermal equivalent diameter but different in hydraulic

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Table 1 Experimental conditions. Cases

Authors

Channel type

p (MPa)

G (kg/m2 s)

q (kW/m2 )

1 2 3

Shitsman (1963) Ornatskiy et al. (1971) Glushchenko and Gandzyuk (1972)

Circular tube Circular tube Annular channel

23.3 25.5 23.5

430 1500 2200

387.33 1810 2410

equivalent diameter shows that heat transfer behavior between the tube and annular channel agrees well. Present study is devoted to numerical simulations of HTD with a well-validated software FLUENT. The purpose of this paper is to numerically study the influence of diameter on HTD in circular tubes and to compare HTD in circular tube with that in annular channel.

The experimental results considered in this study (Shitsman, 1963; Ornatskiy et al., 1971; Glushchenko and Gandzyuk, 1972) are summarized in Table 1. Case 1 and case 2 are in circular tubes at low mass flux and at high mass flux, respectively. Case 3 is at high mass flux in annular channel. 3. Governing equations and turbulence models

∂(vk) ∂(uk) + ∂x ∂y

Continuity: ∂ + ∇ · (U) = 0 ∂t

(1)

Momentum: ∂(U) + ∇ · (U ⊗ U) = ∇ · ( − u ⊗ u) + SM ∂t

(2)



t +

k

∂(vε) ∂(uε) + ∂y ∂x ∂ = ∂x +

A variety of turbulence models are provided in FLUENT, and all models meet the following fundamental governing equations:

8 3 Do = 10

Di = 8

treatment of the boundary layer using the modified Wilcox k–ω model. The general form of low-Reynolds-number k–ε models can be expressed as follows:

∂ = ∂x

2. Experimental data used

D (mm)



t +

ε

 ∂k  ∂x

 ∂ε  ∂x

∂ + ∂y

∂ + ∂y





t +

k

t +

ε

 ∂k  ∂y

+ t G − ε − D

(6)

 ∂ε  ∂y

ε   ε2   c1 f1 t G − c2  f2 + E k k

(7)

  where t = c f  (ε2 /k). Dumping functions (f1 , f2 , f ) and other constants (D and E) are specified in Table 2. The second order upward scheme was used to discretize the turbulence equations to achieve numerical stability in this study. The SIMPLEC scheme was used for coupling the pressure and the velocity fields. The convergence criterion for normalized residual of individual equation was set to be less than 1.0 × 10−6 . The thermal–physical properties were generated using IAPWS-IF97 and were incorporated in FLUENT. 4. Results and discussions 4.1. HTD in circular tube at low mass flux

Energy: ∂(htot ) ∂p + ∇ · (Uhtot ) = ∇ · (∇  − uh) + ∇ · (U · ) − ∂t ∂t + U · SM + SE

(3)

where SM and SE are the source terms, and  is the shear stress. It is assumed that the additional Reynolds stress is associated with mean strain rate in turbulence viscosity models: −u ⊗ u = (∇ U + (∇ U)T ) − (2/3)ı(k + t ∇ · U), and Eddy diffusivity t (t = t /Prt , − uj  = t (∂/∂xj )) is brought in. So momentum and energy equations can be expressed as follows: ∂(U) + ∇ · (U ⊗ U) = B − ∇ p + ∇ · (eff (∇ U + (∇ U)T )) ∂t

Fig. 1 shows wall temperature predicted by various turbulence models. The experimental data was presented by Shitsman (1963) at low mass flux. Under the condition of p = 23.3 MPa, G = 430 kg/m2 s, q = 387.33 kW/m2 , obvious heat transfer deterioration phenomenon with one peak of wall temperature occurs in circular tube of D = 8 mm. Most of the models appear a temperature

(4)

∂p t ∂(htot ) − + ∇ · (Uhtot ) = ∇ · (∇  + ∇ h) + ∇ · (U · ) + SE Prt ∂t ∂t (5) where eff is the effective viscosity defined by eff =  + t and p ¯ · U. is the effective pressure defined by p = p + (2/3)k + (2/3)t ∇ Eight low-Reynolds-number turbulence models have been selected, namely, Abid (AB), Lam–Bremhors (LB), Launder–Sharma (LS), Yang–Shih (YS), Abe–Kondoh–Nagano (AKN), Chang–Hsieh–Chen (CHC), k–ε–v2 –f (V2F) and Shear Stress Transport (SST) model. The SST model combines the robustness of the k–ε model for the bulk flow with the low-Reynolds-number

Fig. 1. Comparison of predictions with Shitsman’s experimental data.

∂y2

0

0

0

0



2 t 

2

(1 − 0.3 exp(−(Ret /6.5) )) · 2 (1 − exp(−y∗ /3.1))

0

∂y

0 0

2

1.90

Fig. 2. Effect of buoyancy force on wall temperature.

1.0

1.0

2

1 − 0.01 exp(−0.0215Rey ) · (1 + 31.66Ret−1.125 ) Chang–Hsieh–Chen (1995) (CHC)

2

exp(1 − (Re /200) )) · (1 − exp(−y /14))

(1 + Abe–Kondoh–Nagano (1994) (AKN)

∗ 2

(1 − exp(−1.5 × 10−4 Rk − 5 × 10−7 Rk3 − 10−10 Rk5 )) Yang–Shih (1993) (YS)

3/4 5/Re

exp(−3.4/(1 + Ret /50) ) Launder–Sharma (1974) (LS)

2

[1 − exp(−0.0165Rey )] × (1 + 20.5/Ret ) Lam–Bremhorst (1981) (LB)

3/4 t

4

Re

1+

 Table 2 Functions and dumping terms in turbulence models.

2

tanh(0.008Rey ) Abid (1993) (AB)

 f Models

1/2

2

0

1 + (0.05/f )

1.0

f1

3

2

(1 − 0.01 exp(Ret2 )) · (1 − 0.0631 exp(−0.0631Rey ))

1.0

1−

0.3 exp(−Ret2 )

1 − exp(−Ret2 )

1−

(1 − exp(−Rey /12))



f2

2 9



exp(1 − Ret2 /36) ·

D

 ∂k1/2 2

0 0

E

+

∂r 2

L. Liu et al. / Nuclear Engineering and Design 255 (2013) 97–104

 ∂ 2 u 2 2 ∂y 2  2  2 t ∂2 u ∂2 v

100

peak in the inlet region, which may be caused by the inlet effect; SST model performs better than other models in predicting the peak which appears in Shitsman’s experiment. Fig. 2 gives predictions using SST model with and without gravity (buoyancy force). It can be seen that the effect of buoyancy force on wall temperature is extremely evident at low mass flux. There is no peak of wall temperature and heat transfer deterioration disappears when buoyancy force is not taken into account in the simulation. So buoyancy force is an important factor to HTD at low mass flux. The effect of tube diameter on heat transfer deterioration at low mass flux was investigated using SST model. Fig. 3 shows variations of wall temperature along the flow direction as tube diameters ranging from 2 to 8 mm, respectively. The comparison was made under the condition of p = 23.3 MPa, G = 430 kg/m2 s, q = 387.33 kW/m2 . There are clearly two kinds of HTD phenomena with two peaks of wall temperature when D = 8 mm: one close to the inlet and the other farther. As the diameter decreases to 4 mm, there is only one peak of wall temperature with the onset of deterioration delayed along the flow direction. As the diameter further decreases to 2 mm, the peak of wall temperature disappears and HTD phenomenon vanishes. In other words, HTD phenomenon becomes more notable with the increase of the tube diameter, which is consistent with the experiment using CO2 (Bae et al., 2010). This is because when the tube diameter increases,

Fig. 3. Variations of wall temperature in circular tubes with different diameters at low mass flux.

L. Liu et al. / Nuclear Engineering and Design 255 (2013) 97–104

Fig. 4. Distribution of turbulence kinetic energy in radial direction in circular tubes with different diameters.

the distribution of radial density is more nonuniform. The natural convection between fluid in boundary layer and mainstream therefore becomes fiercer, which intensifies the dissipation of turbulent kinetic energy (as shown in Fig. 4). So HTD phenomenon is aggravated. As buoyancy force is an important factor to HTD at low mass flux (Fig. 2), the influence of tube diameter on HTD at low mass flux can be elaborated using dimensionless number Bu as follows: Zhou (1983) defined dimensionless number Bu to characterize buoyancy effect: Bu =

Grb 2.7 0.5 Reb Pr b

(8) 3 / 2 ), Grb = (gD3 /b2 )(1 − (/ ¯ b )) = ((b − )gD ¯ b b

where

Reb = ub D/b , and Pr b = cpb /b = b cpb /b . So Bu

= = =

Grb 2.7 0.5 Reb Pr b

(b − )gD3 b b2

·

  2.7  b ub D

·

kb b cpb

0.5 (9)

(b − )gkb0.5 D0.3 b2 u2.7 b0.3 cp b

101

Fig. 5. Comparison of predictions with Ornatskiy’s experimental data.

with deviation from the experimental data to some extent. SST model performs best of all. The effect of tube diameter on heat transfer deterioration at high mass flux is also investigated using SST model as shown in Fig. 6. It can be found that under the condition of p = 25.5 MPa, G = 1500 kg/m2 s, q = 1810 kW/m2 , HTD phenomena are observed in all circular tubes with wall temperature rapidly increasing as the diameters are 3 mm, 5 mm, and 7 mm, respectively. However, the increase of tube diameter aggravates the heat transfer deterioration. For instance, wall temperature in the tube with D = 7 mm is about 100 ◦ C higher than that with D = 3 mm. Therefore, the decrease of the tube diameter can also improve the heat transfer at high mass flux, while such effect is not as strong as that at low mass flux. Researches have indicated that acceleration effect is significantly important for heat transfer deterioration at high mass flux while buoyancy effect at low mass flux (Koshizuka et al., 1995; Palko and Anglart, 2008; Wen and Gu, 2011). The dimensionless number dp+ a is introduced to characterize acceleration effect at high mass flux: dp+ a =

dpa /dz dpa /dz = dptotal /dz (dpa /dz) + (dpf /dz) + (dpg /dz)

(10)

where dpa /dz is the local pressure gradient due to acceleration defined as dpa /dz = G2 (d/dz)(1/); dpf /dz is the local pressure

where b is the bulk fluid density, ¯ is the average fluid density T defined as ¯ = T 1−T T W dT , b is the bulk fluid thermal conducW

b

b

tivity, ub is the bulk fluid velocity, b is the bulk fluid kinematic viscosity, cp is the average heat capacity, and D is the tube diameter. The variables b , kb , ub , b and cp depend on bulk enthalpy and have nothing to do with tube diameter. ¯ decreases with the increase of D, which leads to the enhancement of (b − ) ¯ and D0.3 in expression (9). Consequently, with the increase of D, dimensionless number Bu increases quickly, and buoyancy effect becomes more obvious. So the increase of tube diameter aggravates heat transfer deterioration by intensifying buoyancy effect. 4.2. HTD in circular tube at high mass flux The comparison between wall temperature predicted by eight turbulence models and the experimental data at high mass flux is shown in Fig. 5. The experiment is performed by Ornatskiy et al. (1971) in a circular tube under the condition of p = 25.5 MPa, G = 1500 kg/m2 s, q = 1810 kW/m2 . It can be found that the majority of models can predict the heat transfer deterioration phenomenon,

Fig. 6. Variations of wall temperature in circular tube with different diameters at high mass flux.

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L. Liu et al. / Nuclear Engineering and Design 255 (2013) 97–104

Fig. 7. Comparison of predictions with Glushchenko’s experimental data.

gradient due to friction defined as dpf /dz = Cf · (G2 /2) ¯ · (1/D); dpg /dz is the local pressure gradient due to gravity defined as dpg /dz = (b − )g. The influence of gravity (buoyancy force) on HTD is negligible at high mass flux, so expression (10) can be expressed as follows: dp+ a =

dpa /dz 1 = (dpa /dz) + (dpf /dz) 1 + (dpf /dz)/(dpa /dz)

(11)

For circular tube, the heat balance equation is satisfied: q · D · L = G ·

D2 · h 4

(12)

So h dh 4q = = L GD dz

(13)

As shown, the axial enthalpy gradient dh/dz is inversely proportional to D. Taking expression (13) into the definition of dpa /dz, there is: d dpa = G2 · dz dz

1 

= G2 ·

dh d · dz dh

1 

=

4qG d · D dh

1 

(14)

And dpf /dz can be expressed by Blasius formula: dpf dz

= Cf ·

G2 1 · = 0.3164 · 2¯ D

 GD −0.25 G2 1 

·

2¯

·

D

(15)

Taking expressions (14) and (15) into (11), dp+ a is presented as follows: dp+ a

= =

Fig. 8. Comparison of wall temperature in circular tube with annular channel (inner wall heated).

annular channel with 1 mm gap under the condition of p = 23.5 MPa, G = 2200 kg/m2 s, q = 2410 kW/m2 . Obvious heat transfer deterioration phenomenon occurs with sharp increase of wall temperature at the position of hb = 1650 kJ/kg. All models agree well with the experimental data out of the pseudo-critical region; but within the pseudo-critical region, they do not. All models, except LS model, can predict HTD phenomena with earlier onset of deterioration. After all, SST model agrees better with the experimental data than other models. Variations of wall temperature in circular tube and innerwall-heated annular channel are shown in Fig. 8. The hydraulic equivalent diameters (Dh ) of the two channels are all 2 mm, while the thermal equivalent diameters (DT ) are 2 mm and 4.5 mm respectively. The results manifest that HTD phenomena can be observed in both channels under the condition of p = 23.5 MPa, G = 2200 kg/m2 s, q = 2410 kW/m2 . Despite the same hydraulic equivalent diameter, wall temperature of circular tube still deviates much from that of inner-wall-heated annular channel. HTD phenomenon in circular tube with much higher wall temperature is severer than that in inner-wall-heated annular channel. At the same time, the onset of deterioration in circular tube is earlier. Fig. 9 shows variations of wall temperature in circular tube and double-wall-heated annular channel. With the same Dh (2 mm), DT of the two channels are also the same, i.e. 2 mm. As shown, under the condition of p = 23.5 MPa, G = 2200 kg/m2 s, q = 2410 kW/m2 ,

1 1 + ((0.3164 · (GD/)

−0.25

· (G2 /2) · (1/D))/(4qG/D) · (d/dh)(1/))

1 1 + ((0.31640.25 G0.75 D−0.25 )/(8q)(d/dh)(1/))

According to expression (16), it can be found that dp+ a increases with the increase of D, which results in the more obvious acceleration effect. Consequently, HTD phenomenon is aggravated with quickly rising wall temperature. 4.3. Comparison of HTD in circular tube and annular channel Since the experimental data of HTD in annular channel at low mass flux have not been available yet in the literature, comparison of HTD in circular tube with that in annular channel was just carried out at high mass flux in present study. Fig. 7 shows variations of wall temperature predicted by different turbulence models. The results of LB and YS models are not presented here because of numerical fluctuation. The experimental data is obtained by Glushchenko and Gandzyuk (1972) in the

(16)

there is no obvious difference of wall temperature between the two channels. That is, with the same Dh and DT , the heat transfer deterioration phenomena in circular tube and annular channel are similar to each other. As discussed above, at high mass flux, the variation of wall temperature shows a great difference between circular tube and inner-wall-heated annular channel in spit of the same Dh , and HTD phenomenon in circular tube is severer. The HTD phenomena in circular tube and double-wall-heated annular channel are similar to each other. + The results of dp+ a in all channels are shown in Table 3. dpa has been defined in expression (10). Table 3 summarizes the results of dimensionless number dp+ a in circular tube, inner-wall-heated annular channel and doublewall-heated annular channel at the same bulk enthalpy (hb = 1057,

L. Liu et al. / Nuclear Engineering and Design 255 (2013) 97–104

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Table 3 Comparison of dp+ a in circular tube and annular channel. dpf

dpg dz

dpa dz

dptot dz

dp+ a

hb (kJ/kg)

Channels

1057

Circular tube Annular channel (inner wall heated) Annular channel (both walls heated) (inner wall) Annular channel (both walls heated) (outer wall)

35,709.13 35,899.05 35,329.93 35,480.24

616.0406 656.0448 535.4447 567.5977

193,628.3 3038.518 117,491.1 133,333.7

229,953.5 39,593.61 153,356.5 169,381.5

0.842 0.0767 0.766 0.787

1320

Circular tube Annular channel (inner wall heated) Annular channel (both walls heated) (inner wall) Annular channel (both walls heated) (outer wall)

41,976.3 44,418.82 39,479.34 40,049.34

1251.072 1589.37 868.5183 960.0486

3,120,823 26,518.25 2,956,579 3,049,874

3,164,050 72,526.45 2,996,927 3,090,883

0.986 0.366 0.987 0.987

1348

Circular tube Annular channel (inner wall heated) Annular channel (both walls heated) (inner wall) Annular channel (both walls heated) (outer wall)

44,956.07 49,858.88 40,452.7 41,872.01

1588.866 2140.169 963.3281 1174.998

3,057,545 39,649.49 2,976,665 3,031,732

3,104,090 91,648.54 3,018,082 3,074,779

0.985 0.433 0.987 0.986

1446

Circular tube Annular channel (inner wall heated) Annular channel (both walls heated) (inner wall) Annular channel (both walls heated) (outer wall)

113,078.3 113,466.8 90,972.65 105,485.9

4655.268 4663.026 4122.604 4497.491

598,043.6 82,938.53 1,188,860 773,389.8

715,777.2 201,068.4 1,283,955 883,373.2

0.836 0.412 0.926 0.875

1640

Circular tube Annular channel (inner wall heated) Annular channel (both walls heated) (inner wall) Annular channel (both walls heated) (outer wall)

132,533.4 129,041.9 127,714.5 129,203.4

4220.72 4172.532 4152.928 4174.413

243,026.6 17,039.73 296,445.5 291,432

379,780.6 150,254.2 428,312.9 424,809.8

0.640 0.113 0.692 0.686

dz

(Pa/m)

(Pa/m)

(Pa/m)

(Pa/m)

1320, 1348, 1446 and 1640 kJ/kg). Comparison of dp+ a in different channels are shown in Fig. 10. The local total pressure gradient (dptot /dz) is dominated by dpa /dz in circular tube and doublewall-heated annular channel, especially within the pseudo-critical region (hb = 1320 and 1348 kJ/kg). With little difference in dp+ a , the acceleration effect are almost the same in circular tube and doublewall-heated annular channel, so similar HTD phenomena appear in the two channels. However, dp+ a in circular tube is different from that in inner-wall-heated annular channel, and HTD phenomenon in circular tube is severer with much fiercer acceleration effect. Meanwhile, the much higher wall temperature in circular tube leads to larger radial density gradient, which intensifies the acceleration effect, therefore the heat transfer deterioration is aggravated in circular tube, especially within the pseudo-critical region (Fig. 8). Furthermore, as shown in Figs. 8 and 9, the thermal equivalent diameter has a great influence on heat transfer deterioration, which can be explained by the heat balance equation: q · Ph = G · A ·

dh dz

Fig. 10. Comparison of dp+ a in circular tube with annular channel.

(17)

where Ph is the heating circumference and A is the flowing area.

Thermal equivalent diameter is defined as: DT =

4A Ph

(18)

Taking definition (18) into (17), the expression can be gained as follows: dh 4q = GDT dz

(19)

So dpa /dz can be expressed as follows: dpa d = G2 dz dz

1 

= G2

dh d dz dh

1 

=

4qG d DT dh

1 

(20)

Consequently, the change of DT has an important effect on HTD at high mass flux by changing dpa /dz. This result is consistent with the experiment using CO2 (Bae, 2011). 5. Conclusion

Fig. 9. Comparison of wall temperature in circular tube with annular channel (both walls heated).

Heat transfer deterioration is numerically studied in circular tube and annular channel using eight low-Reynolds-number models including AB, LB, LS, YS, AKN, CHC, V2F and SST to provide

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detailed information on flow and turbulence structure. The main conclusions obtained are as follows: (1) All models considered are able to predict HTD phenomenon, deviating from experimental data to some extent. SST model performs better than other models quantitatively. (2) Acceleration effect is significantly important for heat transfer deterioration at high mass flux while buoyancy effect at low mass flux. The increase of tube diameter leads to the rise of wall temperature and the aggravation of HTD phenomenon, which is more obvious at low mass flux. (3) At high mass flux, HTD phenomenon in circular tube is severer than that in inner-wall-heated annular channel. However, with both walls heated in annular channel, the HTD phenomenon is similar to circular tube. (4) It would be thermal equivalent diameter (DT ) rather than hydraulic equivalent diameter (Dh ) plays an important part in HTD at high mass flux.

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Further reading ANSYS CFX-Solver Theory Guide, 2006. ANSYS, Inc. Canonsburg, PA, USA. Cheng, X., Schulenburg, T., 2001 May. Heat Transfer at Supercritical Pressure. Literature Review and Application to a HPLWR, Wissenschaftliche Berichte FZKA 6609. Forschungszentrum, Karlsruhe. Dobashi, K., et al., 1988. Conceptual design of a high temperature power reactor cooled and moderated by supercritical light water. Ann. Nucl. Energy 25 (8), 487–505. Li, Y.Z., He, W.Z., 2006 October. Engineering Fluid Mechanics. Textbook in Mechanics for Higher Education, vol. 1. Tsinghua University Press, Beijing. Licht, J., 2008. Heat transfer and fluid flow characteristics in supercritical water. Doctoral dissertation, University of Wisconsin-Madison, USA. Lu, Z.Q., 2002 January. Two-phase Flow and Boiling Heat Transfer. Tsinghua University Press, Beijing. Pioro, I.L., Duffey, R.B., 2007. Heat Transfer and Hydraulic Resistance at Supercritical Pressures in Power Engineering Applications. American Society of Mechanical Engineers, New York. Schulenberg, T., et al., 2006. Three Pass Core Design Proposal for a High Performance Light Water Reactor. Proceedings of the Second COEINES International Symposium, November 26-30, 2006, Yokohama, Japan. Yang, S.M., Tao, W.Q., 1998 December. Heat Transfer (Third Edition). Textbook Series for 21st Century. Higher Education Press, Beijing. Wagner, W., Kretzschmar, H., 2008. (International Steam Tables-Properties of Water and Steam Based on the Industrial formulation IAPWS-IF97 (Second Edition)). Springer, Heidelberg, Germany. Wen, Q.L., Gu, H.Y., 2010. Numerical simulation of heat transfer deterioration phenomenon in supercritical water through vertical tube. Ann. Nucl. Energy 37 (May), 1272–1280. Wu, W.Y., 2004. Fluid Mechanics. Peking University Press, Beijing. Zhang, Z.S., Cui, G.X., 2006 September. Fluid Mechanics. Textbook in Mechanics for Higher Education, vol. 2. Tsinghua University Press, Beijing. Zhu, Y., 2010 November. Numerical Investigation of the Flow and Heat Transfer within the Core Cooling Channel of a Supercritical Water Reactor, vol. 8–112. Institut für Kernenergetik und Energiesysteme (IKE 8-112), Universität Stuttgart.