Geometric model

The cooling capacity of the SFP in this study was designed according to 16 MW ^[29], and the size was taken from the actual parameters of the CAP1400, as listed in Table 1. The separated heat pipe radiator is divided into multiple modules that can be flexibly arranged according to the actual needs of the system. As shown in Fig. 2(a), each side of the SFP extends 2 m outward. Separated heat pipes are mounted around the pool, where the evaporators are inserted. Riser and drop tubes are used to connect the condensation segments outside the pool. The spent fuel is placed at the bottom of the pool. Owing to the temperature dissimilarity between the SFP and the evaporation segments, a natural cycle is formed in the pool with the evaporation segment as the cooling source, spent fuel as the heat source, and water as the heat transfer medium. The condensation segment absorbs heat, such that the internal working medium is vaporized into steam under a pressure difference arising from the temperature in the pool. The steam reaches the condensation segment through the riser pipe, and the liquefied working medium flows back to the evaporation segment due to gravity, forming a closed heat transfer process. We arrange the separated heat pipe condensation segment in ambient air, which creates natural circulation with the condensation segment as the heat source and air as the cooling source.

Fig. 2Full-screen View

(Color online) PCS for a spent fuel pool with separated heat pipes: (a) system diagram; (b) numerical model of a single module separated heat pipes with boundary conditions; (c) a separate heat pipe.

Figure 2(b) shows the geometric model of the SFP zone with a single modular separated heat pipes. The condensation segment adopts the form of circular finned tubes to improve the heat transfer efficiency. The evaporation segment is composed of multiple single heat pipes in parallel. Figure 2(c) shows the geometric model of a single separated heat pipe heat exchanger with a ring fin structure in the condensing section of the heat pipe. Table 1 lists the basic dimensions of the SFP and design parameters of the separated heat pipe.

Physical model

2.2.1

Design of the condensation section

Compared with plain, corrugated, and spiral finned tubes, ring finned tubes have a stronger heat exchange effect and can adapt to accident working conditions ^[30]. Therefore, the condensation segment of the separated heat pipe adopts a ring finned tube structure, and the design calculations can be carried out for the air cooler. To obtain the size of the finned tube, we need to estimate the overall heat exchange area and required air volume of the condenser. The heat load is calculated based on condensation of the medium in the tube. Then we assume the total heat transmission coefficient of the condensation segment and estimate the outlet air temperature corresponding to the specific heat formula. Moreover, we need to compute the logarithmic heat transfer temperature difference. Finally, the heat load, total heat transfer coefficient, and heat transfer temperature difference are used to calculate the heat transfer area. Through the above estimation, parameters such as the heat transfer surface area, number of tube bundles, number of tube rows, and air volume of the condenser can be preliminarily determined. The detailed parameters of the condenser structure including the heat transfer and resistance inside and outside the tube bundle can then calculated. Subsequently an appropriate finned tube form is selected, the relevant parameters in the preliminary design are adjusted, and inspection and verification are conducted ^[31]. Fig. 3 illustrates the specific finned condenser design process, and the required design parameters are listed in Table 2.

Table 2Full-screen View

Design parameters of the condensation segment.

Parameter	Value
Heat transfer rate of a single heat pipe, Q (W)	7000
Total heat transfer area, Ac (m2)	2.27
Heat transfer coefficient inside the tube, hi (W/m2·K)	1423
Heat transfer coefficient outside the tube, ho (W/m2·K)	896.4
Intra-pipe scale resistance, Ri (m2·K/W)	0.00008
Fin scale resistance, R∑	-
Contact thermal resistance, Rj (m2·K/W)	0.00007
Pipe wall fouling thermal resistance, Rw (m2·K/W)	0.0069
Overall heat transfer coefficient, kc (W/m2·K)	110.33

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Fig. 3Full-screen View

Design flow chart for the condensation segment.

2.2.3

Thermal resistance network model

Typically, thermal resistance network model is used to forecast the heat transfer properties of a heat pipe. A separate heat pipe comprised multiple heat pipes connected in parallel. Thus, the thermal conductivity of each heat pipe is considered to be identical, and the effective thermal conductivity of the heat pipe can be obtained from the total thermal resistance ^[34]. Figure 4 shows the thermal resistance network model used in this study, ignoring the axial conduction thermal resistance. Table 3 lists the descriptions and calculation formulas for the various thermal resistances of the separated heat pipe, where h is the convective heat transfer coefficient between the fluid and tube wall, $A$ is the convective heat transfer area, the subscripts c, e, l, v are condenser, evaporator, liquid phase, and vapor phase, respectively, $D$ is diameter, $\lambda$ is equivalent thermal conductivity, $H_{\mathrm{fin}}$ is the height of condenser fin, $T_{sat}$ is the saturation temperature of water, $c_{\text{sf}}=0.006$, $h_{fg}$ is specific enthalpy.

Table 3Full-screen View

Calculation equations for the thermal resistances of separated heat pipes ^[34].

Thermal resistance	Description and Formula
Rh,c	Convection thermal resistance of the outer wall of the condenser segment
	$R_{\text{h},\mathrm{\text{c}}}=\frac{T_{\mathrm{\text{c}},\text{out}}-T_{\mathrm{\text{c}},\text{f}}}{Q}=\frac{1}{h_{\text{c}}\cdot A_{\mathrm{\text{c}}}}$
	$h_{\text{c}}=0.1378({\lambda }_{\text{c}}/D_{\text{c}}){({\mathrm{Re}}_{\text{c}})}^{0.718}{\text{Pr}}_{\text{c}}{}^{1/3}{(L_{\text{c}}/H_{\mathrm{\text{fin}}})}^{0.296},{\mathrm{Re}}_{\text{c}}＞2300$
Rw,c	Wall conduction thermal resistance of the condenser
	$R_{\text{w},\mathrm{\text{c}}}=\frac{T_{\mathrm{\text{c}},\mathrm{\text{in}}}-T_{c,out}}{Q}=\frac{\text{ln}(r_{\mathrm{\text{c}},\text{out}}/r_{\text{c},\text{i}\text{n}})}{2\pi {\lambda }_{\text{w}}{L}_{\mathrm{\text{c}}}}$
Rc	Condensation heat resistance of condenser segment
	$R_{\mathrm{\text{c}}}=\frac{T_{\text{sat}}-T_{\text{o}}}{Q}=\frac{1}{h_{\text{c}}\cdot \pi \cdot r_{\text{c},\text{i}\text{n}}^2}$
	$h_{\text{c}}=0.\text{943}{｛\frac{{\rho }_{\text{1}}\cdot \text{g}\cdot {\lambda }_l^3({\rho }_l-{\rho }_{\text{v}})\left[h_{\text{fg}}+0.68{\text{c}}_{\text{p},\mathrm{\text{l}}}(T_{\text{sat}}\text{-}{T}_{\text{o}})\right]}{{\mu }_l{L}_{\text{c}}(T_{\text{sat}}-T_{\text{o}})}｝}^{\text{1/4}}$
Rv	Vapor flow thermal resistance
	-
Rh,e	The convection thermal resistance of the outer wall of the evaporator segment
	$R_{\text{h},\mathrm{\text{e}}}=\frac{T_{\text{e},\text{f}}-T_{\text{e},\text{out}}}{Q}=\frac{1}{h_{\text{e}}\cdot A_{\text{e}}}$
	$h_{\text{e}}=0.11({\lambda }_{\text{e}}/L_{\text{e}}){({\text{Gr}}_{\mathrm{\text{e}}}{\text{Pr}}_{\text{e}})}^{1/3},{\text{Gr}}_{\text{e}}＞2\times 10^{10}$
Rw,e	Convection thermal resistance of the outer wall of the condenser segment
	$R_{\text{w},\mathrm{\text{e}}}=\frac{T_{\text{e},\text{out}}-T_{\text{e},\text{in}}}{Q}=\frac{\text{ln}(r_{\mathrm{\text{e}},\text{out}}/r_{\text{e},\text{in}})}{2\pi {\lambda }_{\text{w}}{L}_{\text{e}}}$
Rb	Boiling heat resistance of the evaporation segment
	$R_{\text{b}}=\frac{T_{\mathrm{\text{s}}}-T_{\text{sat}}}{Q}=\frac{T_{\mathrm{\text{s}}}-T_{\text{sat}}}{q^{\ast }\cdot \pi \cdot r_{\text{e},\text{in}}^2}$
	$q^{\ast }={\mu }_l{h}_{\text{fg}}{[\frac{\text{g}({\rho }_l-{\rho }_{\mathrm{\text{v}}})}{\sigma }]}^{1/2}{(\frac{{\text{c}}_{\text{p},\mathrm{\text{l}}}(T_{\mathrm{\text{sat}}}-T_{\text{s}})}{{\text{c}}_{\text{sf}}{h}_{\text{fg}}{\mathrm{Pr}}_{\mathrm{\text{l}}}^{\text{n}}})}^3$
Rtot	Overall thermal resistance
	$R_{\text{tot}}=R_{\text{h,e}}+{\left[{{\displaystyle {\sum }_{i=1}^n\left(\frac{1}{R_{\text{w},\mathrm{\text{e}}}}\right)}}_i\right]}^{-1}+{\left[{{\displaystyle {\sum }_{i=1}^n\left(\frac{1}{R_{\text{b}}}\right)}}_i\right]}^{-1}+R_{\text{v}}+{\left[{{\displaystyle {\sum }_{j=1}^m\left(\frac{1}{R_{\text{c}}}\right)}}_i\right]}^{-1}+{\left[{{\displaystyle {\sum }_{j=1}^m\left(\frac{1}{R_{\text{w},\text{c}}}\right)}}_i\right]}^{-1}+R_{\text{h},\text{c}}$
λeff	Effective thermal conductivity
	${\lambda }_{\mathrm{\text{eff}}}=\frac{L_{\text{v}}}{R_{\text{tot}}\cdot 2\pi \cdot r_{}^2}(\mathrm{\text{W}}/\text{m}\cdot \text{K})$

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Fig. 4Full-screen View

(Color online) Thermal resistance network model of separated heat pipes.

Decoupling of the three heat transfer processes

Most coupling problems of practical significance cannot be solved analytically; instead, numerical solutions are required. Wang et al. applied porous media to the OpenFOAM solver to calculate the coupled heat transfer in shell–and–tube heat exchangers ^[35]. In addition, RELAP5 is often used to study coupled heat transfer processes in PCS ^[36,37]. Gu et al. ^[38] developed a coupled solution for an Lead/lead–bismuth eutectic (LBE)-cooled reactor using an external loose coupling scheme. This method decomposes the overall model of a pool-type reactor into three models and the output conditions of the former model are the input conditions of the next model. In this study, partition solution and boundary coupling method are used to numerically analyze the heat dissipation characteristics of the separated heat pipe passive cooling system in the SFP. The heat transport in this PCS involves three parts: the SFP, separated heat pipe closed circulation, and atmospheric environment ^[37].

As shown in Fig. 5(a), the condensation segment and atmospheric environment are regarded as region Ⅰ, the separated heat pipe is regarded as region Ⅱ, and the evaporation segment and SFP are regarded as region Ⅲ. Regions Ⅰ and Ⅲ both undergo convective heat transfer, while region Ⅱ experiences coupled heat transfer. This study aims to investigate the influence of varying the parameters in Region Ⅰ on Region Ⅲ using a coupled heat transfer process. The decoupling of the three heat transfer processes is the key point of the numerical simulation in this study.

Fig. 5Full-screen View

The separated heat pipe of PCS: (a) heat transfer processes; (b) decoupling analysis method.

Figure 5(b) shows a flow chart of the decoupling analysis. First, convective heat transfer occurs between the condenser and the ambient air. We adjust the parameters of the condenser fins and the inlet temperature of the condensing working fluid such that the temperature discrepancy between the inlet and outlet of the condensing working fluid is within 5 °C. The closed loop of the working water in the separated heat pipe loop is then calculated to ensure that the loop pressure drop is zero. Subsequently, the natural circulation in the SFP is calculated such that the lift of the water is identical to its flow resistance. Finally, the heat transmission rate of the system is determined to ensure that the total heat transmission capacity is equal to the 16 MW design value.

The natural circulation of the separated heat pipe loop should satisfy the law of energy conservation. In other words, the heat absorption of the evaporation segment is equivalent to the heat dissipation of the condensation segment, and the pressure drop is zero. The pressure drop of the heat pipe loop can be computed using Eqs. (6) and (7): $\Delta P_N=\frac{G_{N1}^2+0.5G_{N2}^2}{2p_i},$ (6) $G_{\text{Ni}}=\frac{W_i}{\frac{\pi }{4}{d}_{\text{Ni}}^2},$ (7) where $G_{\text{Ni}}$ is the mass flow rate of the inlet and outlet, $d_{\text{Ni}}$ is the diameter of the inlet and outlet nozzles, and $W_i$ is the mass flow rate of the fluid in the pipe.

The condition for natural circulation of the SFP is that the buoyancy of water is equal to the flow resistance. The total flow resistance of water in the SFP includes the friction pressure loss and local resistance, which can be calculated using Eqs. (8) and (9). $\Delta P{f}_{\text{w}}={\displaystyle \sum _i\left(f_i\times \frac{L_i}{D_i}\times \frac{\rho u_i^2}{2}\right)}+{\displaystyle \sum _i\left({\xi }_i\times \frac{\rho u_i^2}{2}\right)},$ (8) $f_i=\{\begin{array}{ll}\frac{64}{{\mathrm{Re}}_i},\hfill & {\text{Re}}_i<2300\hfill \\ {\left(1.821\times l_{\text{g}}\left({\mathrm{Re}}_i\right)-1.64\right)}^{-2},\hfill & {\mathrm{Re}}_i\ge 2300\hfill \end{array},$ (9) where $\Delta P{f}_{\text{w}}$ is pressure loss, $f_i$ is resistance coefficient along the path, ${\xi }_i$ is local resistance coefficient.

The decoupling is valid if the heat transmission in the condensation segment is equal to that in the evaporation segment. The formulas for heat transmission are given below: $Q\text{c}=M\cdot \left(i1-i2\right),$ (10) where $M$ denotes the working fluid mass flow rate in the condensation segment, $i1$ is the enthalpy of the inlet water vapor, and $i2$ is the enthalpy of the outlet condensate. $Q\text{e}=A\text{e}\cdot k\cdot \left(t{f}_1-t{f}_2\right),$ (11) $A\text{e}=\frac{Q}{U_0\cdot \Delta T},$ (12) where $Ae$ is the surface area of the evaporation segment, $t_{f1}$ is the temperature of the working fluid in the heat pipe, $t_{f2}$ is the water temperature in the SFP, $\Delta T$ is the average temperature of the heat transfer between the fluid inside and outside the heat pipe, and U₀ is the heat transfer coefficient of the evaporation section, which can be referred to as the thermal resistance network model. The power of the heat source used in the numerical calculations is 105 kW.

Grid independence analysis and model validation

To ensure the accuracy of the numerical simulation, experimental tests were also performed in this study. The test device used in this study represents scale model of the pool. A photograph of the test setup is presented in Fig. 6. The test device includes five sections: an arc heating system, single separated heat pipe radiator, water tank auxiliary heating system, ambient temperature control system, and data acquisition system. The evaporation segment is placed in the pool and the condensation segment is located in the environment. Convective heat transmission in the pool allows heat transfer from evaporation through the riser pipe to the condenser, which dissipates e heat into the air. The condensation segment contains annular fins on the tube bundle and a blower on the upper part. In the tests, the ambient temperature control system provides precise air temperatures, and the arc surface heating system provides different heating powers. A K-type thermocouple is used to measure the temperature at measurement points arranged in separate heat pipes and tanks, and the stability of the test is evaluated by monitoring the temperature changes at each measurement point in the experimental system. Voltage regulators are used to control the power required for each set of trials. During the tests, the output of the portable power regulator is continuously monitored to avoid voltage fluctuations. With this method, we tested the cooling conditions of the condensation segment under natural and forced convection conditions.

Fig. 6Full-screen View

(Color online) Test device diagram.

Grid independence analysis is performed to verify that the number of grids is independent of the final flow calculation results and satisfies the accuracy requirements. The meshes near the heat pipe wall surface and heat source are refined. In the grid independence verification, the heat dissipation of the evaporation segment under different schemes is compared. As shown in Fig. 7(a), the effect of the grid number on the results can be neglected when the grid number exceeds 2.4×10⁶.

Fig. 7Full-screen View

(Color online) Numerical model: (a) grid independence test; (b) comparison of simulation results and experimental data in the condenser; (c) comparison of simulation results and experimental data in the pools.

In this study, a separated heat pipe is adopted as the SFP radiator, which is divided into condensation and evaporation segments. Numerical simulations are then performed for these two models. To improve the model validation accuracy, we assume the other walls of the condenser have adiabatic boundary conditions and that the temperature is the same as the environmental temperature in the experiments. For verification, numerical simulations are performed with heating powers of 3, 4, 6, 7.8, and 8 kW. The uncertainty of the heat pipe heat transfer is ±5.2%, and the uncertainty of the heat source power is ±1.5% ^[39]. When the calculation results are stable, the temperatures at the measurement points arranged in the condensation segment are compared with the simulation results. Then, the temperatures at the measurement points at different heights in the pool are selected for comparison with the simulation results. As shown in Fig. 7(b) and (c), the error between the numerical simulation results and the experimental data is within ±10%, which meets the model verification requirements.

Heat transfer analysis under four seasons climate change

In this study, the SFP heat dissipation unit of an air-cooled heat exchanger is numerically calculated. The thermal properties of the heat sink are analyzed based on the mean temperature of the pool, effective thermal conductivity of the heat pipe, heat dissipation, temperature field, and flow field. Because the temperature distribution of Weihai is consistent, we select the four solar terms of the vernal equinox, summer solstice, autumn equinox, and winter solstice, to reflect the temperature distribution characteristics of the four seasons. Specific data are presented in Table 5.

As shown in Fig. 8, the pool temperature is the highest and heat exchange is the lowest during the summer solstice, while the pool temperature is lowest and heat exchange is highest during the winter solstice. The minimum heat transfer rate reaches 105.384 kW, satisfying the requirements of the single modular heat exchangers (105 kW). As shown in Fig. 9, the water temperature increases gradually with height in areas above 8 m in the pools. At pool heights of 0–8 m, the water temperature gradually decreases with the height because the bottom of the pool is closer to the spent fuel.

Fig. 8Full-screen View

Comparison of the pool heat exchange and average water temperature in different seasons.

Fig. 9Full-screen View

Variation in the average water temperature with height in the XY cross-section in different seasons.

To clearly visualize the water temperature at different heights and flow field in SFP, we selected a region containing only one separated heat pipe heat exchanger for the cloud analysis. As shown in Fig. 10, the water temperature in the SFP is the highest at the summer solstice, and a significant temperature difference exists between the top and bottom of the SFP. Owing to its buoyancy, the heated fluid rises to the upper part of the pool. Consequently, the temperature at the top of the pool increases slightly, and thermal stratification occurs. In addition, the water temperature on the left side of the pool is lower than that on the right side. Fig. 11 shows that the highest velocity in the SFP occurs in the evaporation segment. Three distinct vortex zones are present in the pool, mainly due to the floating force of the natural circulation in the SFP. The water flow at the bottom of the pool rises with a low speed and is distributed in the middle of the SFP owing to buoyancy. Then the water flows along the heat pipe evaporation segment towards the center of the SFP, which causes a temperature discrepancy between the left and right sides of the pool.

Fig. 10Full-screen View

(Color online) Temperature contours of the pool at z = 4.1 m, 6.3 m, 8.5 m, and 10.7 m in different seasons: (a) Vernal equinox, (b) Summer solstice, (c) Autumnal equinox, (d) Winter solstice.

Fig. 11Full-screen View

(Color online) Velocity vector diagram at the ZX section of the SFP: (a) Vernal equinox, (b) Summer solstice, (c) Autumnal equinox, (d) Winter solstice.

Effects of wind speed

Li et al. ^[40] used MM5 and CALMET to simulate the distribution of wind energy in Weihai and analyzed the detailed characteristics of the annual average wind energy distribution in different seasons and simulation periods. The simulation results show that the average yearly wind speed in Weihai ranges from 3.3 m/s to 8.8 m/s. Therefore, we choose wind velocity values of 3.5, 4.5, 5.5, 6.5, 7.5, and 8.5 m/s for the numerical analysis; the air temperature is maintained at 30 ℃. Thus, we explore the performance characteristics of the SFP radiator under different wind speed conditions.

Based on Table 4, we can calculate the effective thermal conductivity under different wind speed conditions. The calculation results show that the effective thermal conductivity of the heat pipe increases continuously with increasing the wind speed. Therefore, we can obtain the function for the heat conductivity of the heat pipes with wind speed by data fitting, as shown in Fig. 12(a): ${\lambda }_{\text{eff}}=-9134.3v^2+191474.9v+1298586.024,$ (13) where ${\lambda }_{\text{eff}}$ is the effective thermal conductivity of the separated heat pipe, and v is the wind speed. The fitted curve shown in Fig. 12(a) is in the form of a parabola with a downward opening. When v is 10.48 m/s, ${\lambda }_{\text{eff}}$ reaches its maximum value. This empirical formula can help improve the prediction of the effect of wind speed on the heat dissipation capacity of PCS. As shown in Fig. 12(b), at any cross-sectional height, the higher the wind speed, the lower the average pool temperature will be, which indicates that the heat pipe radiator removes more heat. As shown in Fig. 12(c), the temperature of the pool gradually decreases with increasing wind speed, and the maximum temperature difference reaches 2.039 K. In addition, the maximum heat dissipation occurs when v is 8 m/s. In conclusion, when v is 8 m/s, the heat pipe radiator achieves optimal heat transmission performance.

Fig. 12Full-screen View

(Color online) Effect of wind speed: (a) fitting curve for the relationship between the heat pipe thermal conductivity and wind speed; (b) variation in the average temperature of the XY cross-section of the SFP with height; (c) pool heat dissipation and average water temperature.

Effect of air temperature

Wang et al. ^[41] used a time series model to study the temperature changes in Weihai City, Shandong Province, over the past 50 years. Based on the average temperature of each season in the model results, the annual temperature fluctuation range in Weihai in a year is -3–30 ℃. Thus, we set the wind speed as 5.5 m/s and the air temperature as 5 ℃, 10 ℃, 15 ℃, 20 ℃, 25 ℃, and 30 ℃ in the simulation conditions. Thus, we also consider the effect of air temperature on the heat transmission properties of the separated heat pipe radiator and natural convection of the pool.

According to the thermal resistance network model, it can be calculated that ${\lambda }_{\text{eff}}$ will decrease steadily with increasing air temperature. Therefore, the heat exchange of the PCS will gradually decrease. The data calculated from Table 4 can be fitted to obtain the relationship between the thermal conductivity of the heat pipe and the wind speed as follows: ${\lambda }_{\text{eff}}=4.9493t_{\text{a}}{}^2-3826.55t_{\text{a}}+\text{2768418}\text{.78,}$ (14) where $t_{\text{a}}$ is the air temperature. Figure 13(a) shows the fitting curve for the relationship between the thermal conductivity of the heat pipe and the air temperature, which is in the form of a parabola with an upward opening. The lower the air temperature, the higher the thermal conductivity of the separated heat tube heat exchanger will be. This curve has specific reference significance for evaluating the performance of the separated heat pipe radiator under different air temperatures.

Fig. 13Full-screen View

Effect of air temperature: (a) Fitting curve for the relationship between the heat pipe thermal conductivity and air temperature; (b) Variation in the average temperature of the XY cross-section of the SFP with height; (c) pool heat dissipation and average water temperature.

As shown in Fig. 13(b), when the air temperature is 10–20 ℃, the average water temperature variation curves on the XY cross-section almost coincide in the height range of 8–12 m. Under these conditions, the difference in the average temperature of the pool is 0.22 ℃, as shown in Fig. 13(c). When the air temperature is 10 ℃, the temperature of the pool is the minimum, and the heat dissipation of the heat pipe radiator is the maximum. When the air temperature is 10 ℃, the heat pipe radiator thus achieves optimal performance.