Heat transfer to supercritical water in a vertical tube with concentrated incident solar heat flux on one side

International Journal of Heat and Mass Transfer 95 (2016) 944–952

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Heat transfer to supercritical water in a vertical tube with concentrated incident solar heat flux on one side G. Zhang, Y. Li ⇑, Y.J. Dai, R.Z. Wang Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e

i n f o

Article history: Received 21 October 2015 Received in revised form 21 December 2015 Accepted 5 January 2016

Keywords: Supercritical water Concentrated incident solar flux Vertical tube Heat transfer

a b s t r a c t A solar tower power plant with supercritical water as a heat-transfer medium in the central receiver is potentially one of the most promising solar thermal power technologies due to its high solar-to-electric efficiency. In this paper, the heat transfer of supercritical water in a vertical tube of a solar tower receiver has been investigated. A 3D mathematical model has been developed to investigate the distribution of heat flux in the circumferential direction of a circular tube heated by concentrated incident solar flux on one side. The RNG k–e model with the standard wall function is employed to describe the turbulent flow of water from a liquid-like state to a gas-like state, and the results are validated with experimental data. For supercritical water in a tube heated on one side by concentrated incident solar heat flux, the maximum wall temperature is located on the sunward outside wall where the incident heat flux is at a maximum. In the pseudo-critical flow region, due to the drastic turbulent diffusion of supercritical water, the temperature distribution of the supercritical water in the tube is evenly distributed at the same flow cross section. To avoid deterioration in the heat transfer of supercritical water, the relation between the threshold incident solar heat flux and mass flux for the supercritical water has been provided. Furthermore, the Nusselt correlation which can be used to predict heat transfer coefficient of the vertical tube heated non-uniformly on one side by concentrated solar flux has been confirmed. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Solar tower power plants receive much attention due to their high solar-to-electric efficiency [1,2] which results from an increasing in the allowable incident heat flux and power cycle efficiency with increasing working temperature. In solar tower power plants, the central receivers have high solar-thermal conversion efficiencies with high acceptable incident heat flux [3]. Indeed, the central receivers, which generally employ either molten nitrate salt or water/steam as the heat transfer fluid, have improved at increasing the acceptable incident heat flux since the 1980s [4]. The receivers using water/steam in Solar One and CESA-1 promise an incident flux of 300 kW/m2; the molten salt solar receiver from Solar Two allows a peak flux of 800 kW/m2. Increasing the allowable incident heat flux is not only helpful for improving the solar-thermal conversion efficiency but also for reducing the Levelized Energy Cost (LEC) from concentrating solar power technologies.

⇑ Corresponding author. Tel./fax: +86 21 34206056. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.01.011 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

Furthermore, the currently operating solar tower power plants mainly operate in subcritical Rankine power cycle with gross thermal-to-electric efficiencies between 30% and 40%. Considering the applications of cycle working fluids with increasing working temperature in coal fired power plants [5] and nuclear power plants [6], the concept of a central receiver with supercritical heat transport fluids [7] has been proposed, and the feasibility of this method has been theoretically demonstrated. The corresponding advantages include high power cycle efficiency, elimination of the steam generator, reduction of costs and so on. In 2014, the world’s first prototype of a solar tower with a supercritical water central receiver successfully produced supercritical steam at a pressure of 23.5 MPa and temperature of 570 °C in Australia. However, further detailed reports are not available. Taking into account the promising applications of supercritical water in solar thermal tower powers in the future, a cavity-type central receiver with supercritical water is proposed and its performance is studied in this paper. Unlike the central receivers with molten nitrate salt or subcritical water/steam, the tubes of a central receiver with supercritical water should have the relatively small diameters due to the high pressure and high temperature [8]. The sunward side of the tubes

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

945

Nomenclature A, B, C cp cp D G g Gr Gr h n q Re r T x, y, z

constant specific heat, J/kg °C average specific heat, J/kg °C diameter, m mass flux, kg/m2 s acceleration of gravity, m2/s Grashof number average Grashof number heat transfer coefficient, kW/m2 K number heat flux, kW/m2 Reynolds number radius of the tube, m temperature, °C axial direction

is used to absorb the concentrated solar energy and transfer the energy to the heat transfer fluid passing the tubes. The incident flux focuses on one side of the receiver tube, and the other side of tube is thermally insulated. Therefore, there is a non-uniform flux distribution on the outside tube surfaces. Generally, the concentrated incident heat flux is regarded to be a uniform and parallel arrangement, a cosine distribution [2] on the sunward side is assumed. The heat transfer to supercritical water in a tube heated by concentrated incident solar heat flux on one side has to be studied to develop a solar tower power with a supercritical water central receiver. Due to the large variations in the thermal–physical properties of supercritical water in the pseudo-critical region, unusual heat transfer behavior (heat transfer enhancement or heat transfer deterioration) occurs in this region. Therefore, extensive experimental and modeling research [8–10] on heat transfer of supercritical water has been conducted since the 1950s, mainly involving supercritical water as coolants in nuclear power plants. These investigations have revealed that the characteristics of heat transfer in supercritical water is strongly affected by heat flux, mass flux, flow direction and buoyancy. Based on these studies, various empirical heat transfer correlations/look-up tables [11,12] have been developed. These empirical methods are well validated within their own applicable ranges and can adequately predict the heat transfer coefficient of supercritical water when buoyancy effects are negligible or slight. In simulations [8,9,13], the reliability of the turbulence model is evaluated to exactly predict the characteristics of heat transfer in supercritical water, including Standard k–e, RNG k–e, SST k–x, Realizable k–e and so on. However, most studies focus on heat transfer in a circular tube heated by uniform heat flux in the circular direction. For the single-side uniform heat flux distribution on the water wall in a supercritical fossil-fired boiler, Li [14] discussed the flow and heat transfer mechanisms using the numerical method. The results revealed that the buoyancy effect is much less significant for single-side uniform heating without quantitative results. Further, the heat flux into the tube in the conventional boiler is uniform based on the one side tube surface. Different from the heat flux for heat exchanger tube in nuclear power plants and the tube used in boilers, the heat flux around the tube is non-uniform in a cosine distribution on the sunward side due to the uniform incident flux and parallel arrangement. To date, to the best knowledge of the authors, there are limited data available for supercritical water in a tube heated on one side by concentrated incident flux. Therefore, it is valuable to investigate the heat transfer behavior of supercritical water in a circular tube heated by incident flux on one side.

Greek symbol circumferential angle, ° dynamic viscosity, Pa s density, kg/m3 average density, kg/m3

a l q q

Subscript ave b in out pc w

and superscript average bulk incident, inside outside pseudo-critica wall

The objective of the present communication is to investigate the characteristics of the heat transfer of supercritical water in a vertical tube heated non-uniformly on one side by concentrated solar flux. To obtain more detail about the distribution of non-uniform heat flux in the circumferential direction of a vertical tube, a 3D simulation is utilized. The RNG k–e model with the standard wall function is employed to carry out numerical simulations. The model is validated by comparing the results to experimental data. The effect of incident flux and mass flux on the heat transfer of supercritical water is studied. Furthermore, the buoyancy criterion, onset of heat transfer deterioration and Nusselt correlation of heat transfer under incident solar heat flux are also investigated in this paper. 2. Modeling 2.1. Physical properties of supercritical water In this article, the inlet pressure of the vertical tube is 24.5 MPa, the bulk temperatures of supercritical water range from 337 to 540 °C. Because the pressure drop inside the circular tube is relatively small compared to the proposed operating pressure for supercritical water in the tube, only the effect of temperature on the supercritical water’s physical properties at 24.5 MPa is considered for the simulation. NIST REFPROP software [15] is applied to calculate the water properties. Fig. 1 gives the property variations of water at 24.5 MPa based on NIST REFPROP. At 24.5 MPa, the pseudo-critical temperature is 383 °C. Near the pseudo-critical line, the special heat shows a sharp peak. The viscosity, density and thermal conductivity decrease dramatically in this region. In addition to the pseudo-critical region, other regions are labeled as the liquid-like region and gas-like region. 2.2. Geometry and distribution of heat flux A cavity-type receiver with a tubular panel is considered for the supercritical water solar tower receiver. Fig. 2 shows the cross section of the calculation model. To simplify the calculation model, a half circular tube is selected as the computational domain in this study. The tubular panel receives the concentrated radiation on the right half and converts it into available thermal energy in the supercritical water, the left half of circular tube is thermally insulated. The tube material is 316L stainless steel. Because the change in the thermal conductivity of 316L stainless steel in the temperature range of 300–600 °C is very small [16], the value of

946

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

280

0.8

330

-5 ×10 10

Temperature 380 430

o

C 480

530

580

100 3

×10 0.8

Cp

80

8

kg/m

3

Cp kJ/(kg·K)

0.6

Pa·s

W/(m·K)

0.6

60

0.4

6

0.4

40

0.2

4

0.2

2

0.0

0.0

553

603

653

703 753 Temperature K

803

20

0

853

Fig. 1. Water properties at 24.5 MPa.

where qin is the available incident heat flux on the circular tube outer surface, and rout is the outside radius of the tube.

y π/2

2.3. Turbulence model

qin

thermal insulation

α π

o

x 0

Fig. 2. The cross section of the calculation model.

24.5 W/(m K) is adopted as the constant in this study. Because a tube with small diameter can achieve more effective heat transfer effect than a large diameter tube near the pseudo-critical region [6,8], an inside diameter of 7.5 mm is almost always used in supercritical water cooled reactors. Therefore, the inside and outside diameters of the tube in this paper are 7.5 and 11 mm, respectively. Under conditions of low incident flux, the corresponding heat transfer coefficient is low, and the pseudo-critical region is long as well. To study the heat transfer near the pseudo-critical region, the calculation continuity of the model near the pseudo-critical region should be ensured. Thus, a tube length of 9 m is considered in this investigation. For a solar tower receiver, the heat loss including conductive, convective and radiative losses can be represented as a percentage (7%) of the overall incident energy on the receiver [17]. Therefore, in this paper, the available incident energy converted into thermal energy in the heat transfer fluid is the overall incident heat flux minus the heat loss. According to the previous literature [18], the incident rays arriving at the surface of the tower receiver are approximately parallel. The heat flux distribution on the circumference of circular tube is non-uniform. This non-uniformity cannot be simply replaced by an equivalent heat flux in the simulation due to the low conductivity of stainless steel. The heat flux distribution on the circumference of the circular tube must be given close to the actual value as possible and is represented through the following equation:

( qðx;y;zÞ ¼


 x qin sin p2  rout if x > 0

0

if x 6 0

ð1Þ

The turbulence model is the most significant factor affecting the accuracy of the numerical simulation of heat transfer in supercritical water. To evaluate the reliability of the turbulence model, various simulations are performed with the turbulence models: Standard k–e, RNG k–e, SST k–e, Realizable k–e and so on [9,10]. For the vertical flow of water in a circular tube, RNG k–e with enhanced wall treatment is more accurate for predicting heat transfer at a supercritical pressure than other models. However, this turbulence model requires a very small height of the first grid off the wall, which makes it unsuitable for 3-D calculations with complex or large geometries. On the other hand, RNG k–e with the standard wall function is also found to have an acceptable prediction capability [10] for heat transfer to supercritical water in a vertical tube. Moreover, the grid requirement of the standard wall function is less strict than that of the enhanced wall treatment. Considering the accuracy of the numerical simulation and computational cost, RNG k–e with standard wall function is employed in this paper. In the wall function approach, the y+ values have generally ranged from 5 to 100 in former studies. Accordingly, in this study, the special mesh partition strategy is adopted as well. Meanwhile, to improve the precision of numerical simulations and accelerate the convergence of flow, a structured mesh is used. 3. Validation and mesh sensitive test 3.1. Validation To verify the accuracy of the turbulence model and simulation method, a comparison of the simulation data with the experimental data obtained by Yamagata et al. [8] has been carried out. The same working conditions are: a tube diameter of 7.5 mm, a pressure of 24.5 MPa, an inlet mass flux of 1260 kg/m2 s and a uniform heat flux of 233 kW/m2. Fig. 3 shows the comparison of the heat transfer coefficient of the simulation and the experiments. It is obviously that the simulated results are in good agreement with the experimental data.

947

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952 Table 1 Operating and initial/boundary conditions of simulations.

70

2

Heat transfer coefficient (kW/m K)

Experiment Calculation

P=24.5 MPa 2 G=1260 kg/m s 2 q=233 kW/m

60 50 40 30

Pressure (MPa)

Inlet temperature (°C)

Incident flux (kW/m2)

Mass flux (kg/m2 s)

1 2 3 4 5 6 7 8

24.5 24.5 24.5 24.5 24.5 24.5 24.5 24.5

337 337 337 337 337 337 337 337

300 400 600 600 600 600 600 800

1000 1000 800 1000 1200 1500 1800 1000

20

ranged from 300 to 800 kW/m2, and the mass flux was from 800 to 2400 kg/m2 s. Table 1 lists the cases studied in this paper.

10 0 320

340

360

380

400

420

440

o

Bulk temperature ( C) Fig. 3. Comparison of the heat transfer coefficient of the simulation and the experiments.

385

380

375

o

Bulk temperature ( C)

Case

Mesh#1: 912000 Mesh#2: 1459200 Mesh#3: 1766400

370

365

360

355 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

z (m) Fig. 4. Mesh sensitivity test.

3.2. Mesh sensitivity test To ensure the calculation accuracy and reduce the computational cost and time, a mesh sensitivity test is performed. A tube length of 1.6 m is used. The incident flux and the mass flux are 1000 kW/m2 and 1500 kg/m2 s, respectively. Fig. 4 gives the mesh sensitivity test for the present study. It can be seen from this picture that the bulk temperatures at different position with different mesh numbers are coincident without obvious differences. The maximum value of the absolute difference is less than 0.33 °C and the corresponding relative difference is no more than 0.1%. This illustrates that the mesh partitions in Fig. 4 are not sensitive in the circular tube calculations. A mesh number of 912,000 for a tube length of 1.6 m is suitable. Therefore, under the same mesh density, an overall mesh number of 5,130,000 for the current calculation model with a length of 9 m is adopted in this paper.

4.1. Heat transfer of supercritical water in a tube heated by incident flux Unlike nuclear reactors or boilers, the vertical tube from a solar tower receiver is heated by incident solar heat flux on one side. The heat flux distribution on the circumference of the circular tube is shown in Eq. (1). The non-uniform distribution of the heat flux also causes an uneven the wall temperature. Fig. 5 shows the temperature distribution of the supercritical water and the tube wall at different regions. The mass flux is 1000 kg/m2 s, and the incident flux is 600 kW/m2. These three figures all show that the maximum wall temperature is located on the outside wall where the incident heat flux is at a maximum. The temperature difference between the front and rear sides is very significant in liquid-like region due to the small thermal conductivity of tube material. In the gas-like region, the relatively small thermal conductivity of supercritical water causes the corresponding temperature difference to be bigger. Therefore, it is important to increase the thermal conductivity of the tube material to reduce the peak wall temperature to maintain the safety of the tube. The non-uniformity of the tube wall temperature causes the temperature distribution of supercritical water to differ from the corresponding temperature distribution under uniform heat flux. Furthermore, the heat transfer characteristics are influenced. The temperature distribution of supercritical water in the pseudo-critical region is basically uniform. This results from the enhancement of turbulent diffusion of supercritical water caused by the flow-acceleration in the pseudo-critical region. In addition, the variation of the corresponding bulk temperature along the vertical tube is given in Fig. 6. It is found that the bulk temperature increase in the pseudo-critical region is very slow compared with that in the liquid-like and gas-like regions. This also means that much heat energy is still required for supercritical water for the transition from the liquid-like phase to gas-like phase, though there is no obvious phase transition. To evaluate the heat transfer of supercritical water in a tube heated by incident flux, the average heat transfer coefficient is employed in the current paper. However, the non-uniformity of the heat flux distribution on the outside of circular tube causes the heat flux distribution on the inside surface of circular tube to be non-uniform. Fig. 7 shows the heat flux distribution on the tube inner wall for different regions. Heat transfer takes places on the entire inner wall, though heat flux is small when a is more than 120°. Therefore, the average heat transfer coefficient used in this study is expressed as:

q ðT in;av e  T b Þ

4. Results and discussion

h¼

Simulations involved in this study were performed at a pressure of 24.5 MPa and an inlet temperature of 337 °C. The incident flux

where q is the incident heat flux, and Tin,ave is the average temperature of tube inside surface. To exactly express Tin,ave, the

ð2Þ

948

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

0

695

750

720

0.002

X

660

715

0

650

-0.002

0.004

-0.002

0.002

0.004

835

865 870

0.002

X

740

0

730

-0.002

5

72 0

72

850

720

-0.004

0

X

820

805 790

775

73 0

-0.004

76 0 0 75

72 5

0 74

c

745 735 725

705 645

-0.004

640

6 35

630

630

715 705

5 69 5 68

5 68

0

66

5 67 0 67 5 66

5 67 5 66 5 65 0 65

b

5

64

63 5

64

a

0.004

Fig. 5. Temperature distribution of supercritical water and the tube wall (a) liquid-like region, (b) pseudo-critical region and (c) gas-like region.

500

1.0

0.8

450 o

Bulk temperature ( C)

liquid-like region pseudo-critical region gas-like region

q/qmax

0.6

400

0.4

350

0.2

0.0

300 0

1

2

3

4

5

6

7

8

9

0

30

60

90 °

120

150

180

Axial location (m) Fig. 7. The heat flux distribution on the inner wall. Fig. 6. Bulk temperature profiles along the vertical tube (G = 1000 kg/m2 s and q = 600 kW/m2).

temperatures of the entire first grid off the inside surface of the tube are averaged, and the corresponding equation is given as follows:

T in;av e ¼

  n 1X ip T in n i¼1 n

ð3Þ

where n is the number of the first grid off the inside surface of the   tube, and T in inp is the corresponding temperature of each grid cell. 4.2. Effect of mass flux and incident flux The effect of the mass flux on the average heat transfer coefficient is given in Fig. 8 for upward, at an incident flux of 600 kW/m2. It is observed that the average heat transfer coefficient depends strongly upon the mass flux, and its value increases with increasing

mass flux. Near the pseudo-critical region, enhanced heat transfer occurs when the mass flux is equal to or greater than 1200 kg/m2 s; meanwhile, a heat transfer deterioration takes place at a mass flux less than 1000 kg/m2 s. The phenomenon of heat transfer deterioration is caused mainly by an increase of the buoyancy effect due to the decrease in mass flux. Therefore, for a certain incident flux, an appropriate mass flux should be adopted to avoid heat transfer deterioration. Fig. 9 shows a comparison of the average heat transfer coefficients under various incident fluxes. The flow direction of supercritical water is upward; the mass flux is 1000 kg/m2 s. It can be seen from Fig. 9 that the peak heat transfer coefficient values decrease with the increase of incident flux and the corresponding bulk temperature shift to the low bulk temperature region, which has been shown in former research to be uniform with the heat transfer of supercritical water in a tube heated by uniform heat

949

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

flux. Near the pseudo-critical region, heat transfer deterioration occurs when the incident fluxes are 600 and 800 kW/m2. The reason is that the increasing incident flux results in the increasing acceleration effect resulting from a sudden drop in density near the pseudo-critical line. In the gas-like region, the heat transfer coefficient decreases with the increase of incident flux. Thus, the outer wall temperature rises rapidly with the increase of incident flux. Fig. 10 gives the variations of the peak temperature of the outer wall with respect to the mass flux under different incident fluxes. According the previous studies on supercritical power techniques [8,19], the outlet temperature of supercritical water is designed to be 540 °C with a pressure of 24.5 MPa in this paper. Due to the limitation of the operating temperature (700 °C) of the tube material, the incident flux should be no more than 600 kW/m2 when the mass flux is greater than 1300 kg/m2 s.

60

G=800 kg/(m s) 2 G=1000 kg/(m s) 2 G=1200 kg/(m s) 2 G=1500 kg/(m s) 2 G=1800 kg/(m s) 2

2

Heat transfer coefficient (kW/m K)

50

40

30

20

10

0 340

350

360

370

380

390

400

o

4.3. Effect of flow direction and buoyancy force

Bulk temperature ( C) Fig. 8. Comparison of heat transfer coefficient with various mass fluxes.

40 2

qin=300 kW/m

2

qin=600 kW/m

2

Heat transfer coefficient (kW/m K)

2

qin=400 kW/m 30

2

qin=800 kW/m 20

10

0 340

350

360

370

380

390

400

o

Bulk temperature ( C) Fig. 9. A comparison of heat transfer coefficient with various incident fluxes.

The variations in the heat transfer coefficient with respect to the bulk temperature for different incident fluxes are presented in Fig. 11 for a mass flux of 1000 kg/m2 s. Upward and downward flows are considered and compared. For an incident flux of 400 kW/m2, the values of the heat transfer coefficient at different bulk temperatures with upward flow and downward flow coincide without evident differences. However, there is a noticeable distinction between the directions of upward and downward around the bulk temperature of 380 °C, for an incident flux of 600 kW/m2. Heat transfer deterioration occurring in the upward flow is not observed in the downward flow. In the opinion of Jackson and Hall [20,21], deterioration begins when the buoyancy forces increase the velocity close to the wall and therefore reduce shear stresses in the bulk. Thus, redistribution reduces turbulence production, which leads to lower heat transfer coefficients. The turbulent diffusivity increases when the flow and the buoyancy for downward flow are in different directions. Therefore, the heat transfer deterioration caused by the buoyancy effect is obviously weakened in downward flow and can even be negligible. However, for the upward flow in a circular tube, it is important to ascertain whether the effect of buoyancy is likely to be significant or not for given conditions. To evaluate the buoyancy effect on supercritical water in a circular tube, a mixed convection parameter defined by the comparison between the buoyancy forces and the inertial forces is proposed by Jackson and Hall [21], and the criterion is given as follows:

800

40

700

650

600

qin=600 kW/m , upward 2 2

qin=600 kW/m , downward

35 2

Heat transfer coefficient (kW/m K)

750

o

Peak temperature of outer wall ( C)

2

400 kW/m 2 500 kW/m 2 600 kW/m 2 700 kW/m

2

qin=400 kW/m , upward 30

qin=400 kW/m , downward 2

25 20 15 10 5

550 900

1200

1500

1800

2100

2

Mass flux (kg/m s) Fig. 10. The peak temperature of outer wall.

2400

0 340

350

360

370

380

390

400

410

420

o

Bulk temperature ( C) Fig. 11. Comparison of heat transfer coefficient for upward and downward flow.

950

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

< 1  105

GD

2

2

q=600 kW/m , G=1000 kg/(m s) 2 2 q=600 kW/m , G=1200 kg/(m s) 2 2 q=400 kW/m , G=600 kg/(m s) 2 2 q=400 kW/m , G=800 kg/(m s)

500 480

o

This criterion characterizes the influence of natural convection on the turbulent vertical flows of supercritical fluids [22]. The buoyancy forces increase the velocity close to the wall and therefore reduce shear stresses in the bulk. Thus redistribution reduces turbulence production. Once the criterion is more than the limiting value of 105, a mixed convection including forced and natural convection appears. The Reynolds number Re is calculated as:

Re ¼

520

ð4Þ

Temperature ( C)

Gr Re2:7

460 440 420

ð5Þ

lb

400

The average Grashof number Gr is expressed as follows:

Gr ¼

380

 Þqb gD3 ðqb  q

l

ð6Þ

2 b

Z

1 ðT in;av e  T b Þ

T in;av e

Tb

qdT

ð7Þ

The limiting value of 105, which is suitable both for the case of uniformly heated on the circumference and the case of uniformly heated on one side, is applied for the onset of mixed convection. When mixed convection takes place, the heat transfer deteriorates in upward flow. However, the criterion is developed on the basis of uniform heat flux, which may unsuitable in the condition of incident flux. A modified criterion is proposed to account for the buoyancy effect in this study. The evolution of the modified parameter Gr=Re2:7 with the dimensionless temperature Tb/Tpc is illustrated in Fig. 12 for incident fluxes of 400 and 600 kW/m2. The tendencies of the parameter Gr=Re2:7 are similar compared with the former results. The ratio value peaks before the bulk temperature reaches the pseudo-critical temperature. Because the onset of mixed convection and the onset of heat transfer deterioration near pseudo-critical region are synchronized, the onset of heat transfer deterioration can be employed to determine the onset of mixed convection. Fig. 13 shows the inside wall temperature distribution along the vertical tube at a of 0. The incident flux is 400 and 600 kW/m2. The abrupt increases in the temperature resulting from heat transfer deterioration are observed for both low mass fluxes. Worse, after the abrupt increases, the growth rates of the

1E-5

2

2

qin=600 kW/m , G=1000 kg/(m s) 2

2

qin=600 kW/m , G=1200 kg/(m s) 2

2

2

2

qin=400 kW/m , G=600 kg/(m s) qin=400 kW/m , G=800 kg/(m s)

1

2

3

4

5

6

Axial location (m) Fig. 13. Variations of the inside wall temperature along vertical tube: a = 0.

where q is the average density, which is expressed as:

q¼

0

Mixed convection

1E-6

temperature increase as well, which cannot facilitate the increase of the output temperature of supercritical water. Combining Fig. 12, the conclusion is confirmed: the modified criterion has the limit of 106 for the condition of incident flux. When the incident flux is equal to the uniform heat flux, the average temperature of the tube inside surface Tin,ave under conditions of the incident flux is smaller than that of the uniform heat flux. Then, under conditions of incident flux, the radial density gradient of the fluid in a circular tube represented by Eq. (7) is reduced, and the corresponding buoyancy forces are reduced as well. Thus, the modified criterion is lower than the criterion proposed for the cases of uniform heat flux by Jackson and Hall. 4.4. Onset of heat transfer deterioration The phenomenon of heat transfer deterioration in supercritical water is mainly caused by flow acceleration as well as buoyancy effects resulting from a sudden drop in density near the pseudocritical region. It occurs for certain combinations of heat flux and mass flux, as shown in Figs. 8 and 9. In addition, it is a slow and smooth behavior; therefore, it is difficult to determine the onset point of heat transfer deterioration. In former works, different definitions were used, most of which are based on the ratio of the heat transfer coefficient (h) to a reference heat transfer coefficient (h0) obtained from the Dittus–Boelter equation. When the ratio of h/h0 equals 0.3, the onset of heat transfer deterioration occurs. Based on experimental data in different diameters vertical tubes, Styrikovich et al. [23] and Yamagata et al. [8] proposed the following relationships between mass flux (G) and heat flux (q) for the onset of heat transfer deterioration:

Gr/Re

2.7

Forced convection 2E-6 1.5E-6 Gr/Re

2.7

1E-7 1E-6

5E-7 0.96

1E-8 0.7

0.8

0.97

0.98 0.99 Tb/Tpc

0.9

1.00

1.0

1.1

1.2

1.3

Tb/Tpc Fig. 12. Evolution of the parameter Gr=Re2:7 with dimensionless temperature Tb/Tpc.

qStyrikovich ¼ 0:58G

ð8Þ

qYamagata ¼ 0:2G1:2

ð9Þ

However, the ratio of 0.3 may be not suitable for the heat transfer of supercritical water in tube heated by incident flux due to the non-uniformity of the temperature distribution on the circumference of circular tube. Therefore, it is essential to reconfirm the criterion of the onset of heat transfer deterioration. Fig. 14 shows the variations of the ratio h/h0 with bulk temperature at a mass flux of 1000 kg/m2 s. The value of the ratio h/h0 reaches a minimum at the pseudo-critical line for all incident fluxes. In Fig. 9, heat transfer deterioration occurs when the incident flux is

951

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

4.5. Nusselt correlation of heat transfer

1.5

It is well known that heat transfer coefficient values are affected by velocity, inside diameter and thermal physical variations. Since the variation of thermal physical properties in the vicinity of the pseudo-critical point is relatively strong, the conventional singlephase correlations cannot accurately predict the heat transfer behaviors of water at supercritical pressures. Extensive empirical heat transfer correlations based on experimental data of supercritical water were developed, among which most of them are a modified form of the Dittus–Boelter equation. Most studies indicate that the heat transfer of supercritical water in a tube is a type of single-phase forced-convection heat transfer. The corresponding Nusselt correlations are given on the basis of following two forms:

1.2

h/h0

0.9

0.6 2

qin=400 kW/m

0.3

2

qin=600 kW/m qin=800 kW/m

2

0.0 340

350

 360

370

380

390

400

qw qb

B2 Nub ¼ CReB1 b Pr b

o

Bulk temperature ( C)



Fig. 14. Variations of the ratio h/h0 with bulk temperature. A2 Nub ¼ CReA1 b Pr b

600 kW/m2. Considering Figs. 9 and 14, a h/h0 ratio of 0.45 should be adopted as the criterion of the onset of heat transfer deterioration. In terms of practical engineering, an abrupt increase in the tube wall temperature resulting from heat transfer deterioration seriously affects the reliability and safe operation of facilities. Therefore, the relations between the threshold incident flux above which the heat transfer deterioration occurs and mass flux for the supercritical water solar tower receiver must be determined. Based on the h/h0 ratio of 0.45, the relations between the threshold incident flux and mass flux for deterioration in heat transfer is given in Fig. 15. According to the equations proposed by Styrikovich and Yamagata, the fitted curves are obtained using the least squares method. The adjusted R-square value of relation 1 and relation 2 are 0.99 and 0.89, respectively. Therefore, the relation 1 is considered for the onset of heat transfer deterioration when supercritical water in a circular tube is heated by incident flux, as shown by the following:

qin ¼ 0:453G

ð10Þ

Compared with the onset of heat transfer deterioration under conditions of uniform heating flux, the onset under conditions of incident flux does not easy occur due to the relatively weak buoyancy effect on heat transfer.

2

Incident radiation flux (kW/m )

800

qw qb

cp cpb

1.2

Relation2: qin=0.106G

600

500

ð12Þ

These correlations were proposed by considering the Reynolds (Re) number, Prandtl (Pr) number and the ratio of the physical properties given at the wall temperature versus at the bulk temperature. Thus, the bulk-fluid temperature might not be the best characteristic temperature at which all thermal physical properties should be evaluated. According to previous studies [12], the average specific heat (cp ), was proposed for use in many supercritical heat-transfer correlations to correct the position where the heat transfer enhancement takes place.

cp ¼

1 ðT in;av e  T b Þ

Z

T in;av e

300

200 1000

ð13Þ

cp dT

Tb

Nevertheless, the Dittus–Boelter equation was used extensively as a basis for various supercritical heat transfer correlations. However, there are few empirical correlations for evaluating the heat transfer coefficient of supercritical water in a circular tube heated by incident flux. In generally, the Nusselt number is calculated by experiments and simulations, the corresponding equation [16] is given as following:

Nub ¼

hr k

ð14Þ



800

ð11Þ

A3

Nub ¼ 0:0043Re0:904 Pr0:684 b b

400

600

B4

where h is the heat transfer coefficient, and k is the heat conductivity coefficient of water. In this study, to predict the heat transfer coefficient of supercritical water in a vertical tube heated by incident flux, an empirical correlation is essential. With minimum discrepancy between the Nusselt number calculated by Eq. (14) and the proposed empirical correlation as the objective, under various conditions in this paper, the following empirical correlation for heat transfer to supercritical water is obtained:

Calculation points Relation1: qin=0.453G

700

B3 

1200

1400

1600

1800

2

Mass flux (kg/(m s)) Fig. 15. Relations between limit incident flux and mass flux for deterioration in heat transfer.

qw qb

0:564 ð15Þ

Fig. 16 gives a comparison between the Nusselt number calculated by Eq. (14) and that predicted by Eq. (15). The points of deterioration are absent. The discrepancies between the Nusselt number calculated by correlation (Eq. (14)) and the Nusselt number predicted by the present model are within 15%. In other word, the Eq. (15) can be applied to predict the heat transfer coefficient of supercritical water in a tube heated by concentrated incident flux on one side.

952

G. Zhang et al. / International Journal of Heat and Mass Transfer 95 (2016) 944–952

(5) The Nusselt correlation for heat transfer to supercritical water in a vertical tube heated by incident flux is given as
0:564 qw Nub ¼ 0:0043Re0:904 Pr0:684 . b b q

Nusselt number calculated by correlation

1000

800

b

+15%

600

Acknowledgment

-15%

This work was supported by the National Key Technology R&D Program under contract No. 2012BAA05B06-6.

400

References 200

0 0

200

400

600

800

1000

Nusselt number predicted by the present model Fig. 16. Comparison between the Nusselt number calculated by Eq. (14) and that predicted by this Eq. (15).

5. Conclusions The characteristics of the heat transfer of supercritical water in a circular tube heated by incident solar heat flux on one side has been investigated using a numerical method. By comparing the non-uniform heating on one side with uniform heating both on the circumference and on one side, the criteria of onset of the mixed convection and the onset of heat transfer deterioration are confirmed. Based on the analysis of the simulated results, some main conclusions can be drawn as following: (1) For the supercritical water in a tube heated by concentrated incident solar heat flux, the maximum wall temperature is located on the outside wall, where the incident heat flux is at a maximum. In the pseudo-critical region, due to the enhancement of turbulent diffusion caused by the flowacceleration of supercritical water, the temperature distribution of supercritical water in tube is basically uniform. (2) Due to a limitation in the operating temperature (700 °C) of the tube material, the incident flux is suggested to be less than 600 kW/m2 when the output temperature of supercritical water is 540 °C. (3) Under conditions of incident flux, the radial density gradient of fluid in a circular tube and the buoyancy forces are reduced because the Tin,ave is lower than that of conditions of uniform heat flux both on the circumference and on one side. Thus, Gr=Re2:7 is modified to a limit of 106 for the onset of mixed convection. (4) When supercritical water in a circular tube is heated by incident flux, a h/h0 ratio of 0.45 is adopted to be the criterion of the onset of deterioration in heat transfer instead of 0.3, which applies to uniform heat flux. In addition, the relation between the threshold incident flux above which the heat transfer deterioration occurs and mass flux for the supercritical water solar tower receiver is given as qin = 0.453G.

[1] O. Behar, A. Khellaf, K. Mohammedi, A review of studies on central receiver solar thermal power plants, Renew. Sustainable Energy Rev. 23 (2013) 12–39. [2] Z. Liao, X. Li, C. Xu, C. Chang, Z. Wang, Allowable flux density on a solar central receiver, Renew. Energy 62 (2014) 747–753. [3] C.K. Ho, B.D. Iverson, Review of high-temperature central receiver designs for concentrating solar power, Renew. Sustainable Energy Rev. 29 (2014) 835– 846. [4] X. Li, W. Kong, Z. Wang, C. Chang, F. Bai, Thermal model and thermodynamic performance of molten salt cavity receiver, Renew. Energy 35 (2010) 981–988. [5] Y. Li, L. Zhou, G. Xu, Y. Fang, S. Zhao, Y. Yang, Thermodynamic analysis and optimization of a double reheat system in an ultra-supercritical power plant, Energy 74 (2014) 202–214. [6] T. Schulenberg, L.K.H. Leung, Y. Oka, Review of R&D for supercritical water cooled reactors, Prog. Nucl. Energy 77 (2014) 282–299. [7] B.D. Kelly, Advanced Thermal Storage for Central Receivers with Supercritical Coolants, Abengoa Sol. Inc., 2010. [8] K. Yamagata, K. Nishikawa, S. Hasegawa, T. Fujii, S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, Int. J. Heat Mass Transfer. (1972) 2575–2593. [9] X. Cheng, B. Kuang, Y.H. Yang, Numerical analysis of heat transfer in supercritical water cooled flow channels, Nucl. Eng. Des. 237 (2007) 240–252. [10] S.H. Kim, Y.I. Kim, Y.Y. Bae, B.H. Cho, Numerical simulation of the vertical upward flow of water in a heated tube at supercritical pressure, Proc. ICAPP, 2004. [11] M.F. Loewenberg, E. Laurien, A. Class, T. Schulenberg, Supercritical water heat transfer in vertical tubes: a look-up table, Prog. Nucl. Energy 50 (2008) 532– 538. [12] S. Mokry, I. Pioro, A. Farah, K. King, S. Gupta, W. Peiman, et al., Development of supercritical water heat-transfer correlation for vertical bare tubes, Nucl. Eng. Des. 241 (2011) 1126–1136. [13] J. Yang, Y. Oka, Y. Ishiwatari, J. Liu, J. Yoo, Numerical investigation of heat transfer in upward flows of supercritical water in circular tubes and tight fuel rod bundles, Nucl. Eng. Des. 237 (2007) 420–430. [14] Z. Li, Y. Wu, J. Lu, D. Zhang, H. Zhang, Heat transfer to supercritical water in circular tubes with circumferentially non-uniform heating, Appl. Therm. Eng. 70 (2014) 190–200. [15] National Institute of Standards Technology, NIST Reference Fluid Thermodynamic and Transport Properties-REFPROP: NIST Standard Reference Database vol. 23 (2007). [16] S. Yang, W. Tao, Heat Transfer, Higher Education Press, 2006 (in Chinese). [17] J.M. Christian, C.K. Ho, CFD simulation and heat loss analysis of the solar two power tower receiver, ASME 2012 6th Int. Conf. Energy Sustain. Collocated ASME 2012 10th Int. Conf. Fuel Cell Sci. Eng. Technol., American Society of Mechanical Engineers, 2012, pp. 227–235. [18] Q. Yu, Z. Wang, E. Xu, X. Li, M. Guo, Modeling and dynamic simulation of the collector and receiver system of 1 MWe DAHAN solar thermal power tower plant, Renew. Energy 43 (2012) 18–29. [19] K. Zhang, Y. Zhang, Y. Guan, D. Zhang, Boiler design for ultra-supercritical coal power plants, Ultra-supercrit. Coal Power Plants, Elsevier, 2013, pp. 104–130. [20] J.D. Jackson, W.B. Hall, Influences of buoyancy on heat transfer to fluids flowing in vertical tubes under turbulent conditions, Turbul. Forced Convect. Channels Bundles, 1979. 613–640. [21] J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical pressure, Turbul. Forced Convect. Channels Bundles, 1979. 563–611. [22] A. Bruch, A. Bontemps, S. Colasson, Experimental investigation of heat transfer of supercritical carbon dioxide flowing in a cooled vertical tube, Int. J. Heat Mass Transfer. 52 (2009) 2589–2598. [23] M.A. Styrikovich, T.K. Margulova, Z.L. Miropol’Skii, Problems in the development of designs of supercritical boilers, Teploenergetika 14 (1967) 4–7.