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Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube
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Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube Xiaohong Hao a,∗, Peixing Xu a,b, Hang Suo a,b, Liejin Guo b a b
School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China State key laboratory of multiphase flow in power engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 28 May 2019 Revised 2 November 2019 Accepted 19 November 2019 Available online xxx Keywords: Supercritical water Water-cooled wall tube Buoyancy effect Half-side heating
a b s t r a c t Flow and heat transfer of supercritical water in the 10 0 0MW supercritical boiler water-cooled wall tube were investigated with numerical simulation method. Half of the water-cooled wall tube was heated near the furnace, and heat flux varied along the height of the furnace. The other half of the water-cooled wall tube was in heat insulation. The RNG k-ε model was used in numerical analysis. The influences of buoyancy, centrifugal force, flow deviation, inlet temperature deviation and specific heat on heat transfer were investigated. Buoyancy effect and reduction of thermal conduction of supercritical water lead to heat transfer deterioration, while centrifugal force results in heat transfer enhancement. The heat transfer coefficient increases with the increase of inlet flow rate. Various heat transfer correlations on predicting half-side heating condition were presented and compared. Furthermore, a new heat transfer correlation was proposed for supercritical water in the water-cooled wall tube under half-side heating with nonuniform heat flux. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In many thermodynamic systems, supercritical water is widely used as an essential working medium because of its high-efficiency convective heat transfer, such as supercritical water-cooled nuclear reactors [1], supercritical coal-fired power plants [2] and hydrogen production in supercritical water [3]. However, the physical properties of supercritical water are special, and it is known that the boundaries between the liquid and gas phase would disappear when the temperature and pressure exceed the critical point. The thermal properties will change dramatically when the temperature of water near the TPC . Therefore, the complicated mechanism of heat transfer of supercritical water still needs investigating. As a result, in many advanced and sophisticated systems, the hydrodynamic characteristics of supercritical water must be investigated thoroughly. Thermal power generation is the main form of power generation in China, and coal is the primary consumption in the thermal power generation. Supercritical boiler units are widely utilized due to their precise advantages in power generation efficiency, carbon dioxide emission reduction and load adjustment, but problems such as water-cooled wall tube bursting are still not solved well.
∗
Corresponding author. E-mail address: [email protected] (X. Hao).
S.W. Liu [4] figured out that the boiler water-wall tube would be bursting on the fire-face side due to oxidation, but this paper analyzes from the perspective of flow and heat transfer of supercritical water in the water-cooled wall tube with numerical simulation method. Besides, the mechanism of tube bursting is investigated. Numerical study of supercritical water was validated against experimental data e.g. by Zhen Zhang [5], Zhi Shen [6] and Jaromin et al. [7]. And simulations and experiments were performed with uniform heat flux along the height of the wall. In this paper, uniform/non-uniform heating means the uniform/nonuniform heat load in circumferential direction, and uniform/nonuniform heat flux means the uniform/non-uniform heat load in height direction. The investigation of Ge Zhang et al. [8] showed buoyancy effect and thermal acceleration effect led to heat transfer deterioration by using the model of vertical tube with uniform heating in circumferential direction. They found that the viscous sublayer characteristics and a great temperature gradient near the wall led to heat transfer deterioration. Mofeng Qu et al. [9] investigated the heat transfer of supercritical water in vertical tube under uniform and non-uniform heating in circumferential direction. Compared with the uniform heating, the occurrence of heat transfer deterioration would be delayed under non-uniform heating. Besides, they found that buoyancy effect was the main reason if heat transfer deteriorates. Zhigang Gao et al. [10] numerically studied heat transfer of supercritical water in horizontal circular tube under non-uniform heating, and the results showed that k-
https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084 0017-9310/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: X. Hao, P. Xu and H. Suo et al., Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084
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ε turbulence model was dependable for numerical simulation and secondary flows resulting from the thermal acceleration and buoyancy would lead to circumferential nonuniformity of heat transfer. Fangbo Li et al. [11] found that heat transfer coefficient under uniform heating was higher than that under half-side heating in the vertical helically-coiled tube. The numerical results showed that the specific heat of supercritical water near the wall under uniform heating was higher than that under half-side heating, and a correlation for heat transfer of supercritical water in the helicallycoiled tube was proposed. Nevertheless, the experiments and simulations mentioned above in papers were performed only with uniform heat flux in height direction, and supercritical water was simulated with a small section of the tube. The main differences between the simulation of flow and heat transfer of supercritical water in the watercooled wall tube and the tube mentioned above are the nonuniform heat flux and the water-cooled wall tube composed of various inclined pipes with different angles. Half-side heating in circumferential direction was utilized in the simulation of Fangbo Li, but the tube model used in this paper is more applicable in boilers. The simulation of the whole water-cooled wall tube can predict the flow and heat transfer of supercritical water well with non-uniform heating in circumferential direction and non-uniform heat flux in height direction. Furthermore, the results can be combined with the actual situation easily, because the simulation is based on the actual working conditions. Now there are many correlations about heat transfer of supercritical water. However, most of them are suitable for uniform heating and uniform heat flux condition. There are no correlations suitable for water-cooled wall tube in boiler. Most of correlations are developed by appending terms which are relevant to inner wall temperature and bulk temperature to the Dittus-Boelter correlation [12]. In this paper, flow and heat transfer of supercritical water in the water-cooled wall tube under non-uniform heat flux and in half-side heating condition are studied with a complicated model, and the influences of flow deviation and inlet temperature deviation are further discussed. At last, a new heat transfer correlation is proposed. 2. Physical and numerical model
In order to study the flow and heat transfer of supercritical water, the numerical investigation is conducted. The water-cooled wall tube with SA-213T23 material surrounds the furnace and is shaped like a spring (Fig. 3), and the water-cooled wall and three-view drawing of the water-cooled wall tube are shown in the Fig. 1. The relationship between physical parameters of SA213T23 material and temperature shown in the following Eq. (1) is used. The inner diameter is 24 mm, and the outer diameter is 38.1 mm.
⎧ λ = 4.64481 × 10−9 · T3 − 2.79581 × 10−5 · T2 + 2.13315 ⎪ ⎪ −2 ⎨ · T + 32.6853
c = 4.16667 × 10−7 · T3 − 6.66437 × 10−4 · T2 + 0.529145 · T ⎪ ⎪ ⎩P
h=
q Tw − Tb
(2)
where q is local heat flux, Tw is local inner wall temperature, Tb is local bulk temperature. The Nusselt number is calculated by the Eq. (3)
Nu =
hl
(3)
λ
where h is calcualted by Eq. (2), l is characteristic length, λ is thermal conductivity 2.3. Numerical method 2.3.1. Governing equations In the paper, the flow and heat transfer of supercritical water can be regarded as a steady state process, and the continuum medium model is utilized to simulate the flow and heat transfer of supercritical water in the water-cooled wall tube. Besides, the fluid mechanics is constrained by three governing equations: continuity equation, momentum equation and energy equation [13]. Continuity equation:
→ ∇ · ρ− v =0
(4)
Momentum equation:
→− − → 2 ∂u ∂ ∂ ui ∂ uj → ∇ · ρ− v→ v = −∇ p + ρ − g + + − μ k δij + F μ ∂xj ∂ xj ∂ xi 3 ∂ xk (5) − → where F contains other model-dependent source Energy equation:
− ∇· → v (ρ E + p) = −∇ · h j J j + sh
2.1. Physical model
×10
of supercritical water. And heat flux distribution is imported into the Fluent by using user-defined functions. The total height of the water-cooled wall tube is 64.725 m, and the length of the tube is 136 m. The inlet and outlet boundary conditions are respectively set to the velocity-inlet and pressure-outlet. The boundary and operating condition are listed in Table.1. According to the Eq. (2), the heat transfer coefficient is calculated.
(1)
+ 331.696
2.2. Boundary condition The surface of the tube near the furnace side is endothermic and the surface of the tube far from the furnace side is adiabatic as the FigS. 2 and 3 show. Based on the measured data recorded from the actual operating conditions of the power plant, the heat flux varied with height of the tube is shown in the Fig. 4. The ANSYS Fluent code is utilized to simulate the flow and heat transfer
(6)
j
where sh is turbulent diffusion flux of enthalpy Correlation between temperature and enthalpy:
H = c p T − c pre f Tre f
(7)
The Reynolds-Averaged Navier–Stokes is currently recognized and widely used among the turbulent numerical simulation methods. Its core idea is to solve Eq. (5) instead of directly solving the instantaneous Navier–Strokes equation. 2.3.2. Turbulence model With the development of computer technology, k-ε model, kτ model and k-ω model are widely used for numerical simulation. Sharabi et al. [14] compared 8 models, including k-ε model, k-τ model and k-ω model. The comparison showed that k-ε model provided more reliable results than other near wall treatments (kτ model, k-ω model, wall functions). Some observations with kε model could be considered reliable enough. K-τ model and k-ω model couldn’t predict the heat transfer deterioration well under low heat flux. With the increase of heat flux, these models could predict the occurrence of heat transfer deterioration, but it underestimated the severity of heat transfer deterioration. The RNG k-ε model and SST k-ω model show great agreement to experimental
Please cite this article as: X. Hao, P. Xu and H. Suo et al., Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084
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Fig. 1. Water-cooled wall and three-view drawing of the water-cooled wall tube. Table.1 Boundary and operating conditions. Operation pressure, MPa 30.42
Mass flow rate, kg/m2 •s 2207
Inlet temperature, K 604
Heat flux Fig. 4
Fig. 2. Cross-sectional drawing of tube.
Fig. 4. Heat flux of the water-cooled wall tube.
Fig. 3. The schematic diagram of the water-cooled wall tube.
enough, but simulation with RNG k-ε model is more accurate than with SST k-ω model and RSM model in a semi-heated tube. Besides, Fangbo Li et al. [11] investigated supercritical water numerically in vertical helically-coiled tube by using RNG k-ε model. The results showed that the heat transfer coefficient under half-side heating is lower than that under uniform heating. This shows RNG k-ε model has a certain accuracy in the bends. Mofeng Qu et al. [9] noted that SST k-ω model was not accurate enough to predict the wall temperature well if heat transfer deterioration occurred. Considering all these factors, the RNG k-ε model is chosen to simulate the flow and heat transfer of supercritical water in the watercooled wall tube. 2.4. Meshes
data [15–17] in flow and heat transfer of supercritical fluid. Junhua Bai et al. [18] applied SST k-ω model, RNG k-e model and RSM model to investigate the accuracy of turbulence model for simulation. The results showed that simulation with RNG k-e model, SST k-ω model and RSM model with uniform heating is accurate
The inner diameter is 24 mm, and the outer diameter is 38.1 mm. The total height of the water-cooled wall tube is 64.725 m, and the length of tube is 136 m. And the meshes are divided by the means of densifying radial grid meshes and reduc-
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Fig. 5. Computational meshes.
3. Results and discussion 3.1. Temperature distribution
Fig. 6. Comparison between experimental data and numerical data.
ing axial grid meshes, because the length of the water-cooled wall tube is long, the hydraulic diameter is very small relative to the length, and hydrodynamic characteristics do not change dramatically in the length direction. The physical model is divided into structure grid meshes and hexahedron meshes are generated. Furthermore, grid meshes are densified in the length direction in order to ensure the accurate change of the hydrodynamic characteristics of supercritical water at the turn. Y+ is lower than 30, thickness of the first mesh is 75 × 10−6 m and mesh size of each layer increases by 1.3 times. The phenomenon that the temperature of outlet decreases is inconsistent with the actual situation when the grid number is 54.94 million. When the grid number is 65.08 million, the temperature distribution tends to be stable with the increase of the grid number. With the calculation accuracy and computation time considered, the total number of the mesh is 77.68 million in the computational domain, as the Fig. 5 shows.
3.1.1. Temperature distribution of inner surface of tube The Fig. 7 shows the temperature distribution of the inner surface of the water-cooled wall tube under pressure of 30.42 MPa and mass flux of 2207 kg m−2 s−1 . It is shown that the temperature distribution of inner surface of tube at circumferential angles of 180°(1), 150°(2), 120°(3), 90°(4), 60°(5), 30°(6), 0°(7) have roughly the same trend. Near the pseudo-critical region (bulk enthalpy is 2209 kJ kg−1 ), the temperature growth rate of supercritical water is slowed down obviously due to the local maximum of specific heat value and the rapid decrease of density. Curve 5,6,7 and curve 1,2,3 show almost the same temperature distribution respectively. The temperature difference between different curves begins to augment from the entrance of the tube and decreases near the pseudo-critical point. Regardless of the inlet temperature deviation or the inlet flow deviation, the increase of fluid temperature will decrease in the pseudo-critical region (near 676.4 K). When the inlet flow rate increases, the difference between temperature at circumferential angles will decrease in the second half of the pipe. And this will enhance the heat transfer from the inner wall to the supercritical water. Therefore, the increase of inlet flow rate will increase the heat transfer coefficient. 3.1.2. Temperature distribution of outer surface of tube The Fig. 8 shows the outer wall temperature distribution under the pressure of 30.42 MPa and the mass flux of 2207 kg m−2 s−1 . Wall temperatures at different circumferential angles change little from 70 m to 90 m not only because the temperature is near the pseudo-critical point, but also because the heat flux of the watercooled wall tube is at local minimum. At the outlet, the specific heat of T23 as material is raising because of the increasing temperature. As the Fig. 4 explicated, heat flux of tube becomes low. This leads to temperature reduction at outlet section. The temperature reduction at heating side will happen whether the inlet temperature deviation or the inlet flow deviation occurred or not.
2.5. Model validation The result of the experiment is utilized to verify the RNG k-ε model. Fig. 6 shows the comparison of the inner wall temperature between numerical data by utilizing RNG k-ε model and experimental data from Mofeng Qu et al. [19] to validate the proposed model, and it shows a great agreement between the numerical data and experimental data.
3.1.3. Circumferential temperature distribution In order to better understand the influence of non-uniform heat flux on heat transfer, the circumferential temperature distribution of the inner wall and the outer wall for various bulk enthalpy is shown in Fig. 9. In Fig. 9(a), the temperature distribution is even when the enthalpy changes from 20 0 0 to 250 0 kJ kg−1 , and according to the Fig. 14, heat transfer coefficient is close
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Fig. 7. Temperature distribution of inner surface versus bulk enthalpy at different inlet flow rate and temperature.
5
Fig. 8. Temperature distribution of outer surface versus length of tube at different inlet flow rate and temperature.
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Fig. 10. Two cross sections at the turning.
to the maximum. The heat from the wall can be quickly carried away by the supercritical water. Apart from circumferential temperature of supercritical water in 1500 kJ kg−1 having little change because of the low heat flux, the outer wall temperature increases along the circumferential angles when the bulk enthalpies are 1745 kJ kg−1 and 2751 kJ kg−1 (from 60° to 120°). In Fig. 9(b), the outer wall temperature with enthalpy of 2751 kJ kg−1 is lower than with enthalpies of 2001 kJ kg−1 , 2249 kJ kg−1 and 2503 kJ kg−1 along the circumferential angles (from 90° to 180°) because of the low heat flux. In Fig. 9(c), it is clear that the temperature differences between outer and inner wall along the circumferential angles are 7.52°C and 20 °C when the enthalpies are 1500 kJ kg−1 and 2751 kJ kg−1 , but the temperature differences are more than 51°C on the heating side when enthalpies are 1745 kJ kg−1 , 2001 kJ kg−1 and 2249 kJ kg−1 . As a result, the fluid temperature and wall temperature will not reach the maximum at the same time.
3.2. Effect of secondary flow for supercritical water under half-side heating with non-uniform heat flux in the water-cooled wall tube
Fig. 9. Temperature distribution of inner wall and outer wall versus circumferential angle at different enthalpies.
When the supercritical water flows in the water-cooled wall tube, it will pass through various turnings. And as shown in the Fig. 10, the fluid passes through two cross section A and B, and the length of arrow stands for the secondary flow rate. The average temperatures are 662.4 K(before the pseudo-critical temperature), 677.2 K(near the pseudo-critical temperature), and 689.1 K(after the pseudo-critical temperature) respectively. In Fig. 11, vectors show the direction and velocity magnitude of secondary flow, and different colors represent the total velocity magnitude of the flow. The six cross sections are labeled as A(662.4 K), B(662.4 K), A(677.2 K), B(677.2 K), A(689.1 K), B(689.1 K). Secondary flow is the flow perpendicular to the main axis, and the influence of secondary flow in the turning is discussed. When the fluid just enters the turning (cross section A), the horizontal flow direction is biased upwards due to the buoyancy effect. In addition, the axial flow velocity peak shifts to the non-heating side, and the peak region is more concentrated than cross section B’s. Before the exit of turning (cross section B), the secondary flow exacerbates the mixing of the fluid and the temperature gradient of the fluid boundary layer on the heating side becomes larger. As a result, local heat transfer enhancement occurs. It can also be seen from the length of the arrow that the secondary flow rate in cross section B is higher than that in cross section A.
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Fig. 11. Flow field velocity distribution at different cross sections.
Fig. 12. Temperature distribution at different cross sections.
For the temperature distribution of the cross section A(662.4 K) and B(662.4 K), the temperature of fluid does not reach the pseudo-critical point. The axial flow velocity and secondary flow velocity are low, and the heat on the heating side can not spread well. So the radial temperature of fluid at the cross section A(662.4) and B(662.4) changes greatly. The temperature of fluid at the cross section A(677.2 K) and B(677.2 K) is near the pseudo-critical point. The value of specific heat is at maximum, and fluid flow rate starts to increase as the temperature increases and the density decreases. This will lead to the temperature variation of the cross section being the smallest in these cross sections as the Fig. 12 explicated. The temperature variation of fluid at the cross section A(689.1 K) and B(689.1 K) starts to increase when the fluid passes through the large specific heat region. The Fig. 13 shows the temperature distribution of the secondary flow of supercritical water with respect to the different cross sections. The horizontal axis means the distance from the inner wall surface of the heating side to the non-heating side. Two vortices distributed up and down will be generated because of the secondary flow at the turning. These two vortex directions are opposite. Therefore, there are two points where the secondary flow velocity is almost at zero. The secondary flow velocity of the B cross section is always larger than the A cross section in different turnings. When the supercritical water just enters the turning, the cen-
Fig. 13. Secondary flow velocity distribution.
trifugal force has little effect. In the second half of the turning, the direction of the fluid flow velocity is almost at 90° to the entrance velocity direction. Supercritical water interacts with the wall surface and centrifugal force enhances. This leads to heat transfer en-
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Fig. 14. Heat transfer coefficient and Bo∗ versus bulk enthalpy at different circumferential angles.
hancement. As the flow rate increases with temperature, the secondary flow rate increases. Simultaneously, it can be found in the secondary flow velocity distribution that the secondary flow velocity on the heating side is more intense than on the non-heating side because of the great temperature change on the heating side. 3.3. Heat transfer characteristics and effect of buoyancy The Fig. 14 shows the heat transfer coefficient distribution along the different circumferential angles under pressure of 30.42 MPa and mass flux of 2207 kg m−2 s−1 . The heat transfer coefficient increases significantly at the beginning, reaches the maximum near the pseudo-critical point, and declines after the pseudo-critical point. We found that the peak values of heat transfer coefficients are approximately the same but heat transfer coefficient reaches the peak at different enthalpies, such as at 2187 kJ kg−1 in curve d and at 2307 kJ kg−1 in curve a. At enthalpy ranging from 20 0 0 to 2100 kJ kg−1 , there is a region where heat transfer coefficient drops dramatically. That is the length of the tube is about 50– 60 m and the height is about 25 m when the enthalpy is from 20 0 0 to 210 0 kJ kg−1 . And it can be concluded that the local areas absorb more heat and the wall temperature rises rapidly in Figs. 4 and 8. Heat transfer deterioration has occurred in this region. According to the thermophysical properties of supercritical water, thermal conductivity declines when the enthalpy is from 20 0 0 to 210 0 kJ kg−1 (666.8 K–672 K), and the temperature gradient of the fluid near the wall decreases gradually. As a result, the heat cannot be transferred to the mainstream. Besides, buoyancy effect is the main reason of heat transfer deterioration. Buoyancy caused by radical density gradient has great influence on heat transfer of supercritical water. Jackson et al. [20] proposed the Eq. (8) in order to quantify the effect of buoyancy. Gao et al. [10] have verified that the Eq. (8) can predict the buoyancy effect in horizontal flow successfully. Heat transfer coefficients and Bo∗ against the bulk enthalpy of the fluid are depicted in Fig. 14. The Bo∗ increases continuously in low enthalpy, and the rapid change of thermal properties leads to increasing heat transfer. The Bo∗ gets to the maximum at 20 0 0 kJ kg−1 , and the heat transfer coefficient reaches the local minimum at the same enthalpy. The equation of Bo∗ can predict the heat transfer deterioration well. Then the Bo∗ decreases in high enthalpy and it is below 10−5 at last because Reynold number raises due to the declining density and viscosity and increasing velocity. It means the Jackson–Hall rule of Bo∗
Fig. 15. Relative deviations of the correlations versus bulk enthalpy.
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Fig. 16. Comparison between the Nu in calculation and simulation.
<10−5 is effective to predict buoyancy effect and heat transfer deterioration. ∗
Bo =
Grb
(8)
Re2b.7
3.4. Heat transfer correlation for supercritical water under half-side heating with non-uniform heat flux in water-cooled wall tube Fig. 15 shows the relative deviations of four heat transfer correlations(Dittus-Boelter correlation [12], Yamagata correlation [21], Mao correlation [22], Li correlation [11]) of supercritical water and correlations utilized to predict Nu of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux. These heat transfer correlations are shown as follows: The Dittus-Boelter correlation:
Nu = 0.023 · Re0b.8 Pr0b.4
(9)
The Yamagata correlation:
Nu = 0.0135 · Re0b.85 Pr0b.8 Fc
(10)
where Fc is calculated by
⎧ Fc = 1forE > 1 ⎪ ⎪ n1 ⎪ ⎪ .05 ⎪ Fc = 0.067Pr−0 c p /cpb for0 ≤ E ≤ 1, ⎪ pc ⎪ ⎪ n2 ⎨ Fc = c p /cpb
forE < 0,
⎪ E = (Tpc − Tb )/(TW − Tb ), ⎪ ⎪ ⎪ ⎪n = −0.77 1 + 1/Pr + 1.49, ⎪ ( pc ) ⎪ 1 ⎪ ⎩ n2 = 1.44(1 + 1/Prpc ) − 0.53
(11)
Nu = 0.0161 ·
(ρw /ρb )0.851
(12)
(13)
(ρw /ρb )0.545 λw /λb 0.218
(14)
And the relative deviations of Nu are calculated. The relative deviation of Yamagata correlation is too great when the enthalpy is around 2209 kJ kg−1 (pseudo-critical region), and it is more than 60%. Besides, yamagata correlation exhibits high fit at the non-pseudo-critical region. The relative deviations of DittusBoelter correlation and Mao correlation have opposite trends. The prediction of Dittus-Boelter correlation is higher than actual data (relative deviation is more than 20%). On the contrary, the Mao’s is lower (relative deviation is less than −20%). The performance of Li’s correlation is better than other three correlations. Heat transfer coefficient is not only related to Re, Pr, thermal conductivity(λ) and density(ρ ), but also related to dynamic viscosity(μ). Dynamic viscosity affects the velocity of supercritical water, especially near the wall. And velocity has a great impact on heat transfer. Therefore, the dynamic viscosity term also needs to be considered when fitting the correlation. A new correlation for the calculation of heat transfer coefficients of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is proposed. But thermal conductivity term is omitted because supercritical water mainly exchanges heat in a convective manner when it flows in the water-cooled wall tube. The parameters are: p = 30.42 MPa, qw = 0–400 kW m−2 , d = 24 mm, G = 2207 kg m−2 s−1 . 0.6127
P RE − CAL RD = CAL
where
Prb = c p μb /λb , c p = (Hw − Hb )/(Tw − Tb )
0.469
Nu = 0.0289 · Re0b.805 Prb
Nu = 0.01225 · Re0b.8356 Prb
The Mao correlation: 0.632 Re0b.848 Prb
The Li correlation:
(ρw /ρb )0.2127 μw /μb
(15) (16)
where PRE is the result given by correlations, CAL is the numerical result, RD is relative deviation, MRD is mean relative deviation and MARD is mean absolute relative deviation.
MRD =
n 1 RDi n
(17)
i=1
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MARD =
n 1 |RDi | n
(18)
i=1
Fig. 16 shows the comparison between the predicted Nu with Eq. (15) and the data. Almost all of relative deviations of predicted Nu are less than ±20%, and most relative deviations are less than ±10%. Mean relative deviation is about 2.9%, and mean absolute relative deviation is about 9%. The correlation can predict heat transfer well when the boiler is at 100% BMCR(Boiler Maximum Continuous Rating: p = 30.42 MPa, G = 2207 kg m−2 s−1 , qw = 0– 400 kW m−2 , d = 24 mm). 4. Conclusions In this paper, numerical simulation on flow and heat transfer of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is investigated. And main conclusions achieved can be summarized as follows: •
•
•
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The difference between temperature at circumferential angles will decrease in the second half of the pipe and heat transfer coefficient enhances when the inlet flow rate increases. In the outlet section, low heat flux will result in the phenomenon of temperature reduction. And this phenomenon will always happen whether the inlet temperature deviation or inlet flow deviation occurred or not. The fluid temperature and wall temperature will not reach the maximum at the same time. The axial flow direction will be biased upwards and axial flow velocity peak shifts to the non-heating side because of the buoyancy effect when the fluid just enters the turning. And local heat transfer enhances before the exit of turning. In different circumferential angles, the heat transfer coefficients reach the peak at different enthalpies, but the values of peak heat transfer coefficients are almost the same. Before the pseudo-critical point, there is a heat transfer deterioration region. According to the Jackson–Hall rule of Bo∗ <10−5 , it shows a great accuracy for supercritical water in the water-cooled wall tube, and it explains why this heat transfer deterioration region exists before the pseudo-critical temperature. Apart from the buoyancy effect, the decrease of thermal conductivity of supercritical water is the other reason for heat transfer deterioration. A new correlation for supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is proposed and compared with the existing correlations, which shows greater accuracy and superiority than previous correlations.
Declaration of Competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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Acknowledgements The authors gratefully acknowledge the high performance computing center of University of Shanghai for Science and Technology for providing software and hardware support and the Open Fund of State key laboratoryfor providing support in this study.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube Xiaohong Hao a,∗, Peixing Xu a,b, Hang Suo a,b, Liejin Guo b a b
School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China State key laboratory of multiphase flow in power engineering, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
i n f o
Article history: Received 28 May 2019 Revised 2 November 2019 Accepted 19 November 2019 Available online xxx Keywords: Supercritical water Water-cooled wall tube Buoyancy effect Half-side heating
a b s t r a c t Flow and heat transfer of supercritical water in the 10 0 0MW supercritical boiler water-cooled wall tube were investigated with numerical simulation method. Half of the water-cooled wall tube was heated near the furnace, and heat flux varied along the height of the furnace. The other half of the water-cooled wall tube was in heat insulation. The RNG k-ε model was used in numerical analysis. The influences of buoyancy, centrifugal force, flow deviation, inlet temperature deviation and specific heat on heat transfer were investigated. Buoyancy effect and reduction of thermal conduction of supercritical water lead to heat transfer deterioration, while centrifugal force results in heat transfer enhancement. The heat transfer coefficient increases with the increase of inlet flow rate. Various heat transfer correlations on predicting half-side heating condition were presented and compared. Furthermore, a new heat transfer correlation was proposed for supercritical water in the water-cooled wall tube under half-side heating with nonuniform heat flux. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In many thermodynamic systems, supercritical water is widely used as an essential working medium because of its high-efficiency convective heat transfer, such as supercritical water-cooled nuclear reactors [1], supercritical coal-fired power plants [2] and hydrogen production in supercritical water [3]. However, the physical properties of supercritical water are special, and it is known that the boundaries between the liquid and gas phase would disappear when the temperature and pressure exceed the critical point. The thermal properties will change dramatically when the temperature of water near the TPC . Therefore, the complicated mechanism of heat transfer of supercritical water still needs investigating. As a result, in many advanced and sophisticated systems, the hydrodynamic characteristics of supercritical water must be investigated thoroughly. Thermal power generation is the main form of power generation in China, and coal is the primary consumption in the thermal power generation. Supercritical boiler units are widely utilized due to their precise advantages in power generation efficiency, carbon dioxide emission reduction and load adjustment, but problems such as water-cooled wall tube bursting are still not solved well.
∗
Corresponding author. E-mail address: [email protected] (X. Hao).
S.W. Liu [4] figured out that the boiler water-wall tube would be bursting on the fire-face side due to oxidation, but this paper analyzes from the perspective of flow and heat transfer of supercritical water in the water-cooled wall tube with numerical simulation method. Besides, the mechanism of tube bursting is investigated. Numerical study of supercritical water was validated against experimental data e.g. by Zhen Zhang [5], Zhi Shen [6] and Jaromin et al. [7]. And simulations and experiments were performed with uniform heat flux along the height of the wall. In this paper, uniform/non-uniform heating means the uniform/nonuniform heat load in circumferential direction, and uniform/nonuniform heat flux means the uniform/non-uniform heat load in height direction. The investigation of Ge Zhang et al. [8] showed buoyancy effect and thermal acceleration effect led to heat transfer deterioration by using the model of vertical tube with uniform heating in circumferential direction. They found that the viscous sublayer characteristics and a great temperature gradient near the wall led to heat transfer deterioration. Mofeng Qu et al. [9] investigated the heat transfer of supercritical water in vertical tube under uniform and non-uniform heating in circumferential direction. Compared with the uniform heating, the occurrence of heat transfer deterioration would be delayed under non-uniform heating. Besides, they found that buoyancy effect was the main reason if heat transfer deteriorates. Zhigang Gao et al. [10] numerically studied heat transfer of supercritical water in horizontal circular tube under non-uniform heating, and the results showed that k-
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ε turbulence model was dependable for numerical simulation and secondary flows resulting from the thermal acceleration and buoyancy would lead to circumferential nonuniformity of heat transfer. Fangbo Li et al. [11] found that heat transfer coefficient under uniform heating was higher than that under half-side heating in the vertical helically-coiled tube. The numerical results showed that the specific heat of supercritical water near the wall under uniform heating was higher than that under half-side heating, and a correlation for heat transfer of supercritical water in the helicallycoiled tube was proposed. Nevertheless, the experiments and simulations mentioned above in papers were performed only with uniform heat flux in height direction, and supercritical water was simulated with a small section of the tube. The main differences between the simulation of flow and heat transfer of supercritical water in the watercooled wall tube and the tube mentioned above are the nonuniform heat flux and the water-cooled wall tube composed of various inclined pipes with different angles. Half-side heating in circumferential direction was utilized in the simulation of Fangbo Li, but the tube model used in this paper is more applicable in boilers. The simulation of the whole water-cooled wall tube can predict the flow and heat transfer of supercritical water well with non-uniform heating in circumferential direction and non-uniform heat flux in height direction. Furthermore, the results can be combined with the actual situation easily, because the simulation is based on the actual working conditions. Now there are many correlations about heat transfer of supercritical water. However, most of them are suitable for uniform heating and uniform heat flux condition. There are no correlations suitable for water-cooled wall tube in boiler. Most of correlations are developed by appending terms which are relevant to inner wall temperature and bulk temperature to the Dittus-Boelter correlation [12]. In this paper, flow and heat transfer of supercritical water in the water-cooled wall tube under non-uniform heat flux and in half-side heating condition are studied with a complicated model, and the influences of flow deviation and inlet temperature deviation are further discussed. At last, a new heat transfer correlation is proposed. 2. Physical and numerical model
In order to study the flow and heat transfer of supercritical water, the numerical investigation is conducted. The water-cooled wall tube with SA-213T23 material surrounds the furnace and is shaped like a spring (Fig. 3), and the water-cooled wall and three-view drawing of the water-cooled wall tube are shown in the Fig. 1. The relationship between physical parameters of SA213T23 material and temperature shown in the following Eq. (1) is used. The inner diameter is 24 mm, and the outer diameter is 38.1 mm.
⎧ λ = 4.64481 × 10−9 · T3 − 2.79581 × 10−5 · T2 + 2.13315 ⎪ ⎪ −2 ⎨ · T + 32.6853
c = 4.16667 × 10−7 · T3 − 6.66437 × 10−4 · T2 + 0.529145 · T ⎪ ⎪ ⎩P
h=
q Tw − Tb
(2)
where q is local heat flux, Tw is local inner wall temperature, Tb is local bulk temperature. The Nusselt number is calculated by the Eq. (3)
Nu =
hl
(3)
λ
where h is calcualted by Eq. (2), l is characteristic length, λ is thermal conductivity 2.3. Numerical method 2.3.1. Governing equations In the paper, the flow and heat transfer of supercritical water can be regarded as a steady state process, and the continuum medium model is utilized to simulate the flow and heat transfer of supercritical water in the water-cooled wall tube. Besides, the fluid mechanics is constrained by three governing equations: continuity equation, momentum equation and energy equation [13]. Continuity equation:
→ ∇ · ρ− v =0
(4)
Momentum equation:
→− − → 2 ∂u ∂ ∂ ui ∂ uj → ∇ · ρ− v→ v = −∇ p + ρ − g + + − μ k δij + F μ ∂xj ∂ xj ∂ xi 3 ∂ xk (5) − → where F contains other model-dependent source Energy equation:
− ∇· → v (ρ E + p) = −∇ · h j J j + sh
2.1. Physical model
×10
of supercritical water. And heat flux distribution is imported into the Fluent by using user-defined functions. The total height of the water-cooled wall tube is 64.725 m, and the length of the tube is 136 m. The inlet and outlet boundary conditions are respectively set to the velocity-inlet and pressure-outlet. The boundary and operating condition are listed in Table.1. According to the Eq. (2), the heat transfer coefficient is calculated.
(1)
+ 331.696
2.2. Boundary condition The surface of the tube near the furnace side is endothermic and the surface of the tube far from the furnace side is adiabatic as the FigS. 2 and 3 show. Based on the measured data recorded from the actual operating conditions of the power plant, the heat flux varied with height of the tube is shown in the Fig. 4. The ANSYS Fluent code is utilized to simulate the flow and heat transfer
(6)
j
where sh is turbulent diffusion flux of enthalpy Correlation between temperature and enthalpy:
H = c p T − c pre f Tre f
(7)
The Reynolds-Averaged Navier–Stokes is currently recognized and widely used among the turbulent numerical simulation methods. Its core idea is to solve Eq. (5) instead of directly solving the instantaneous Navier–Strokes equation. 2.3.2. Turbulence model With the development of computer technology, k-ε model, kτ model and k-ω model are widely used for numerical simulation. Sharabi et al. [14] compared 8 models, including k-ε model, k-τ model and k-ω model. The comparison showed that k-ε model provided more reliable results than other near wall treatments (kτ model, k-ω model, wall functions). Some observations with kε model could be considered reliable enough. K-τ model and k-ω model couldn’t predict the heat transfer deterioration well under low heat flux. With the increase of heat flux, these models could predict the occurrence of heat transfer deterioration, but it underestimated the severity of heat transfer deterioration. The RNG k-ε model and SST k-ω model show great agreement to experimental
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Fig. 1. Water-cooled wall and three-view drawing of the water-cooled wall tube. Table.1 Boundary and operating conditions. Operation pressure, MPa 30.42
Mass flow rate, kg/m2 •s 2207
Inlet temperature, K 604
Heat flux Fig. 4
Fig. 2. Cross-sectional drawing of tube.
Fig. 4. Heat flux of the water-cooled wall tube.
Fig. 3. The schematic diagram of the water-cooled wall tube.
enough, but simulation with RNG k-ε model is more accurate than with SST k-ω model and RSM model in a semi-heated tube. Besides, Fangbo Li et al. [11] investigated supercritical water numerically in vertical helically-coiled tube by using RNG k-ε model. The results showed that the heat transfer coefficient under half-side heating is lower than that under uniform heating. This shows RNG k-ε model has a certain accuracy in the bends. Mofeng Qu et al. [9] noted that SST k-ω model was not accurate enough to predict the wall temperature well if heat transfer deterioration occurred. Considering all these factors, the RNG k-ε model is chosen to simulate the flow and heat transfer of supercritical water in the watercooled wall tube. 2.4. Meshes
data [15–17] in flow and heat transfer of supercritical fluid. Junhua Bai et al. [18] applied SST k-ω model, RNG k-e model and RSM model to investigate the accuracy of turbulence model for simulation. The results showed that simulation with RNG k-e model, SST k-ω model and RSM model with uniform heating is accurate
The inner diameter is 24 mm, and the outer diameter is 38.1 mm. The total height of the water-cooled wall tube is 64.725 m, and the length of tube is 136 m. And the meshes are divided by the means of densifying radial grid meshes and reduc-
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Fig. 5. Computational meshes.
3. Results and discussion 3.1. Temperature distribution
Fig. 6. Comparison between experimental data and numerical data.
ing axial grid meshes, because the length of the water-cooled wall tube is long, the hydraulic diameter is very small relative to the length, and hydrodynamic characteristics do not change dramatically in the length direction. The physical model is divided into structure grid meshes and hexahedron meshes are generated. Furthermore, grid meshes are densified in the length direction in order to ensure the accurate change of the hydrodynamic characteristics of supercritical water at the turn. Y+ is lower than 30, thickness of the first mesh is 75 × 10−6 m and mesh size of each layer increases by 1.3 times. The phenomenon that the temperature of outlet decreases is inconsistent with the actual situation when the grid number is 54.94 million. When the grid number is 65.08 million, the temperature distribution tends to be stable with the increase of the grid number. With the calculation accuracy and computation time considered, the total number of the mesh is 77.68 million in the computational domain, as the Fig. 5 shows.
3.1.1. Temperature distribution of inner surface of tube The Fig. 7 shows the temperature distribution of the inner surface of the water-cooled wall tube under pressure of 30.42 MPa and mass flux of 2207 kg m−2 s−1 . It is shown that the temperature distribution of inner surface of tube at circumferential angles of 180°(1), 150°(2), 120°(3), 90°(4), 60°(5), 30°(6), 0°(7) have roughly the same trend. Near the pseudo-critical region (bulk enthalpy is 2209 kJ kg−1 ), the temperature growth rate of supercritical water is slowed down obviously due to the local maximum of specific heat value and the rapid decrease of density. Curve 5,6,7 and curve 1,2,3 show almost the same temperature distribution respectively. The temperature difference between different curves begins to augment from the entrance of the tube and decreases near the pseudo-critical point. Regardless of the inlet temperature deviation or the inlet flow deviation, the increase of fluid temperature will decrease in the pseudo-critical region (near 676.4 K). When the inlet flow rate increases, the difference between temperature at circumferential angles will decrease in the second half of the pipe. And this will enhance the heat transfer from the inner wall to the supercritical water. Therefore, the increase of inlet flow rate will increase the heat transfer coefficient. 3.1.2. Temperature distribution of outer surface of tube The Fig. 8 shows the outer wall temperature distribution under the pressure of 30.42 MPa and the mass flux of 2207 kg m−2 s−1 . Wall temperatures at different circumferential angles change little from 70 m to 90 m not only because the temperature is near the pseudo-critical point, but also because the heat flux of the watercooled wall tube is at local minimum. At the outlet, the specific heat of T23 as material is raising because of the increasing temperature. As the Fig. 4 explicated, heat flux of tube becomes low. This leads to temperature reduction at outlet section. The temperature reduction at heating side will happen whether the inlet temperature deviation or the inlet flow deviation occurred or not.
2.5. Model validation The result of the experiment is utilized to verify the RNG k-ε model. Fig. 6 shows the comparison of the inner wall temperature between numerical data by utilizing RNG k-ε model and experimental data from Mofeng Qu et al. [19] to validate the proposed model, and it shows a great agreement between the numerical data and experimental data.
3.1.3. Circumferential temperature distribution In order to better understand the influence of non-uniform heat flux on heat transfer, the circumferential temperature distribution of the inner wall and the outer wall for various bulk enthalpy is shown in Fig. 9. In Fig. 9(a), the temperature distribution is even when the enthalpy changes from 20 0 0 to 250 0 kJ kg−1 , and according to the Fig. 14, heat transfer coefficient is close
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Fig. 7. Temperature distribution of inner surface versus bulk enthalpy at different inlet flow rate and temperature.
5
Fig. 8. Temperature distribution of outer surface versus length of tube at different inlet flow rate and temperature.
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Fig. 10. Two cross sections at the turning.
to the maximum. The heat from the wall can be quickly carried away by the supercritical water. Apart from circumferential temperature of supercritical water in 1500 kJ kg−1 having little change because of the low heat flux, the outer wall temperature increases along the circumferential angles when the bulk enthalpies are 1745 kJ kg−1 and 2751 kJ kg−1 (from 60° to 120°). In Fig. 9(b), the outer wall temperature with enthalpy of 2751 kJ kg−1 is lower than with enthalpies of 2001 kJ kg−1 , 2249 kJ kg−1 and 2503 kJ kg−1 along the circumferential angles (from 90° to 180°) because of the low heat flux. In Fig. 9(c), it is clear that the temperature differences between outer and inner wall along the circumferential angles are 7.52°C and 20 °C when the enthalpies are 1500 kJ kg−1 and 2751 kJ kg−1 , but the temperature differences are more than 51°C on the heating side when enthalpies are 1745 kJ kg−1 , 2001 kJ kg−1 and 2249 kJ kg−1 . As a result, the fluid temperature and wall temperature will not reach the maximum at the same time.
3.2. Effect of secondary flow for supercritical water under half-side heating with non-uniform heat flux in the water-cooled wall tube
Fig. 9. Temperature distribution of inner wall and outer wall versus circumferential angle at different enthalpies.
When the supercritical water flows in the water-cooled wall tube, it will pass through various turnings. And as shown in the Fig. 10, the fluid passes through two cross section A and B, and the length of arrow stands for the secondary flow rate. The average temperatures are 662.4 K(before the pseudo-critical temperature), 677.2 K(near the pseudo-critical temperature), and 689.1 K(after the pseudo-critical temperature) respectively. In Fig. 11, vectors show the direction and velocity magnitude of secondary flow, and different colors represent the total velocity magnitude of the flow. The six cross sections are labeled as A(662.4 K), B(662.4 K), A(677.2 K), B(677.2 K), A(689.1 K), B(689.1 K). Secondary flow is the flow perpendicular to the main axis, and the influence of secondary flow in the turning is discussed. When the fluid just enters the turning (cross section A), the horizontal flow direction is biased upwards due to the buoyancy effect. In addition, the axial flow velocity peak shifts to the non-heating side, and the peak region is more concentrated than cross section B’s. Before the exit of turning (cross section B), the secondary flow exacerbates the mixing of the fluid and the temperature gradient of the fluid boundary layer on the heating side becomes larger. As a result, local heat transfer enhancement occurs. It can also be seen from the length of the arrow that the secondary flow rate in cross section B is higher than that in cross section A.
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Fig. 11. Flow field velocity distribution at different cross sections.
Fig. 12. Temperature distribution at different cross sections.
For the temperature distribution of the cross section A(662.4 K) and B(662.4 K), the temperature of fluid does not reach the pseudo-critical point. The axial flow velocity and secondary flow velocity are low, and the heat on the heating side can not spread well. So the radial temperature of fluid at the cross section A(662.4) and B(662.4) changes greatly. The temperature of fluid at the cross section A(677.2 K) and B(677.2 K) is near the pseudo-critical point. The value of specific heat is at maximum, and fluid flow rate starts to increase as the temperature increases and the density decreases. This will lead to the temperature variation of the cross section being the smallest in these cross sections as the Fig. 12 explicated. The temperature variation of fluid at the cross section A(689.1 K) and B(689.1 K) starts to increase when the fluid passes through the large specific heat region. The Fig. 13 shows the temperature distribution of the secondary flow of supercritical water with respect to the different cross sections. The horizontal axis means the distance from the inner wall surface of the heating side to the non-heating side. Two vortices distributed up and down will be generated because of the secondary flow at the turning. These two vortex directions are opposite. Therefore, there are two points where the secondary flow velocity is almost at zero. The secondary flow velocity of the B cross section is always larger than the A cross section in different turnings. When the supercritical water just enters the turning, the cen-
Fig. 13. Secondary flow velocity distribution.
trifugal force has little effect. In the second half of the turning, the direction of the fluid flow velocity is almost at 90° to the entrance velocity direction. Supercritical water interacts with the wall surface and centrifugal force enhances. This leads to heat transfer en-
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Fig. 14. Heat transfer coefficient and Bo∗ versus bulk enthalpy at different circumferential angles.
hancement. As the flow rate increases with temperature, the secondary flow rate increases. Simultaneously, it can be found in the secondary flow velocity distribution that the secondary flow velocity on the heating side is more intense than on the non-heating side because of the great temperature change on the heating side. 3.3. Heat transfer characteristics and effect of buoyancy The Fig. 14 shows the heat transfer coefficient distribution along the different circumferential angles under pressure of 30.42 MPa and mass flux of 2207 kg m−2 s−1 . The heat transfer coefficient increases significantly at the beginning, reaches the maximum near the pseudo-critical point, and declines after the pseudo-critical point. We found that the peak values of heat transfer coefficients are approximately the same but heat transfer coefficient reaches the peak at different enthalpies, such as at 2187 kJ kg−1 in curve d and at 2307 kJ kg−1 in curve a. At enthalpy ranging from 20 0 0 to 2100 kJ kg−1 , there is a region where heat transfer coefficient drops dramatically. That is the length of the tube is about 50– 60 m and the height is about 25 m when the enthalpy is from 20 0 0 to 210 0 kJ kg−1 . And it can be concluded that the local areas absorb more heat and the wall temperature rises rapidly in Figs. 4 and 8. Heat transfer deterioration has occurred in this region. According to the thermophysical properties of supercritical water, thermal conductivity declines when the enthalpy is from 20 0 0 to 210 0 kJ kg−1 (666.8 K–672 K), and the temperature gradient of the fluid near the wall decreases gradually. As a result, the heat cannot be transferred to the mainstream. Besides, buoyancy effect is the main reason of heat transfer deterioration. Buoyancy caused by radical density gradient has great influence on heat transfer of supercritical water. Jackson et al. [20] proposed the Eq. (8) in order to quantify the effect of buoyancy. Gao et al. [10] have verified that the Eq. (8) can predict the buoyancy effect in horizontal flow successfully. Heat transfer coefficients and Bo∗ against the bulk enthalpy of the fluid are depicted in Fig. 14. The Bo∗ increases continuously in low enthalpy, and the rapid change of thermal properties leads to increasing heat transfer. The Bo∗ gets to the maximum at 20 0 0 kJ kg−1 , and the heat transfer coefficient reaches the local minimum at the same enthalpy. The equation of Bo∗ can predict the heat transfer deterioration well. Then the Bo∗ decreases in high enthalpy and it is below 10−5 at last because Reynold number raises due to the declining density and viscosity and increasing velocity. It means the Jackson–Hall rule of Bo∗
Fig. 15. Relative deviations of the correlations versus bulk enthalpy.
Please cite this article as: X. Hao, P. Xu and H. Suo et al., Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084
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9
Fig. 16. Comparison between the Nu in calculation and simulation.
<10−5 is effective to predict buoyancy effect and heat transfer deterioration. ∗
Bo =
Grb
(8)
Re2b.7
3.4. Heat transfer correlation for supercritical water under half-side heating with non-uniform heat flux in water-cooled wall tube Fig. 15 shows the relative deviations of four heat transfer correlations(Dittus-Boelter correlation [12], Yamagata correlation [21], Mao correlation [22], Li correlation [11]) of supercritical water and correlations utilized to predict Nu of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux. These heat transfer correlations are shown as follows: The Dittus-Boelter correlation:
Nu = 0.023 · Re0b.8 Pr0b.4
(9)
The Yamagata correlation:
Nu = 0.0135 · Re0b.85 Pr0b.8 Fc
(10)
where Fc is calculated by
⎧ Fc = 1forE > 1 ⎪ ⎪ n1 ⎪ ⎪ .05 ⎪ Fc = 0.067Pr−0 c p /cpb for0 ≤ E ≤ 1, ⎪ pc ⎪ ⎪ n2 ⎨ Fc = c p /cpb
forE < 0,
⎪ E = (Tpc − Tb )/(TW − Tb ), ⎪ ⎪ ⎪ ⎪n = −0.77 1 + 1/Pr + 1.49, ⎪ ( pc ) ⎪ 1 ⎪ ⎩ n2 = 1.44(1 + 1/Prpc ) − 0.53
(11)
Nu = 0.0161 ·
(ρw /ρb )0.851
(12)
(13)
(ρw /ρb )0.545 λw /λb 0.218
(14)
And the relative deviations of Nu are calculated. The relative deviation of Yamagata correlation is too great when the enthalpy is around 2209 kJ kg−1 (pseudo-critical region), and it is more than 60%. Besides, yamagata correlation exhibits high fit at the non-pseudo-critical region. The relative deviations of DittusBoelter correlation and Mao correlation have opposite trends. The prediction of Dittus-Boelter correlation is higher than actual data (relative deviation is more than 20%). On the contrary, the Mao’s is lower (relative deviation is less than −20%). The performance of Li’s correlation is better than other three correlations. Heat transfer coefficient is not only related to Re, Pr, thermal conductivity(λ) and density(ρ ), but also related to dynamic viscosity(μ). Dynamic viscosity affects the velocity of supercritical water, especially near the wall. And velocity has a great impact on heat transfer. Therefore, the dynamic viscosity term also needs to be considered when fitting the correlation. A new correlation for the calculation of heat transfer coefficients of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is proposed. But thermal conductivity term is omitted because supercritical water mainly exchanges heat in a convective manner when it flows in the water-cooled wall tube. The parameters are: p = 30.42 MPa, qw = 0–400 kW m−2 , d = 24 mm, G = 2207 kg m−2 s−1 . 0.6127
P RE − CAL RD = CAL
where
Prb = c p μb /λb , c p = (Hw − Hb )/(Tw − Tb )
0.469
Nu = 0.0289 · Re0b.805 Prb
Nu = 0.01225 · Re0b.8356 Prb
The Mao correlation: 0.632 Re0b.848 Prb
The Li correlation:
(ρw /ρb )0.2127 μw /μb
(15) (16)
where PRE is the result given by correlations, CAL is the numerical result, RD is relative deviation, MRD is mean relative deviation and MARD is mean absolute relative deviation.
MRD =
n 1 RDi n
(17)
i=1
Please cite this article as: X. Hao, P. Xu and H. Suo et al., Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084
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MARD =
n 1 |RDi | n
(18)
i=1
Fig. 16 shows the comparison between the predicted Nu with Eq. (15) and the data. Almost all of relative deviations of predicted Nu are less than ±20%, and most relative deviations are less than ±10%. Mean relative deviation is about 2.9%, and mean absolute relative deviation is about 9%. The correlation can predict heat transfer well when the boiler is at 100% BMCR(Boiler Maximum Continuous Rating: p = 30.42 MPa, G = 2207 kg m−2 s−1 , qw = 0– 400 kW m−2 , d = 24 mm). 4. Conclusions In this paper, numerical simulation on flow and heat transfer of supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is investigated. And main conclusions achieved can be summarized as follows: •
•
•
•
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The difference between temperature at circumferential angles will decrease in the second half of the pipe and heat transfer coefficient enhances when the inlet flow rate increases. In the outlet section, low heat flux will result in the phenomenon of temperature reduction. And this phenomenon will always happen whether the inlet temperature deviation or inlet flow deviation occurred or not. The fluid temperature and wall temperature will not reach the maximum at the same time. The axial flow direction will be biased upwards and axial flow velocity peak shifts to the non-heating side because of the buoyancy effect when the fluid just enters the turning. And local heat transfer enhances before the exit of turning. In different circumferential angles, the heat transfer coefficients reach the peak at different enthalpies, but the values of peak heat transfer coefficients are almost the same. Before the pseudo-critical point, there is a heat transfer deterioration region. According to the Jackson–Hall rule of Bo∗ <10−5 , it shows a great accuracy for supercritical water in the water-cooled wall tube, and it explains why this heat transfer deterioration region exists before the pseudo-critical temperature. Apart from the buoyancy effect, the decrease of thermal conductivity of supercritical water is the other reason for heat transfer deterioration. A new correlation for supercritical water in the water-cooled wall tube under half-side heating with non-uniform heat flux is proposed and compared with the existing correlations, which shows greater accuracy and superiority than previous correlations.
Declaration of Competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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Acknowledgements The authors gratefully acknowledge the high performance computing center of University of Shanghai for Science and Technology for providing software and hardware support and the Open Fund of State key laboratoryfor providing support in this study.
Please cite this article as: X. Hao, P. Xu and H. Suo et al., Numerical investigation of flow and heat transfer of supercritical water in the water-cooled wall tube, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119084