Numerical investigation of natural convection characteristics of a heat pipe-cooled passive residual heat removal system for molten salt reactors

NUCLEAR ENERGY SCIENCE AND ENGINEERING

Numerical investigation of natural convection characteristics of a heat pipe-cooled passive residual heat removal system for molten salt reactors

Cheng-Long Wang ，

Hao Qin，

Da-Lin Zhang，

Wen-Xi Tian，

Guang-Hui Su，

Sui-Zheng Qiu

Nuclear Science and Techniques

Vol.31, No.7

Article number 65

Published in print 01 Jul 2020

Available online 18 Jun 2020

DOI：10.1007/s41365-020-00780-z

40300

The limited availability of studies on the natural convection heat transfer characteristics of fluoride salt has hindered progress in the design of passive residual heat removal systems (PRHRS) for molten salt reactors. This paper presents results from a numerical investigation of natural convection heat transfer characteristics of fluoride salt and heat pipes in the drain tank of a PRHRS. Simulation results are compared with experimental data, demonstrating the accuracy of the calculation methodology. Temperature distribution of fluoride salt and heat transfer characteristics are obtained and analyzed. The radial temperature of liquid fluoride salt in the drain tank shows a uniform distribution, while temperatures increase with increasing axial height from the bottom to the top of the drain tank. In addition, natural convection intensity increases with increasing height of the heat pipes in the tank. Spacing between heat pipes has no obvious effect on the natural convection heat transfer coefficient. This study will contribute to the design of passive heat removal systems for advanced nuclear reactors.

Molten salt reactorPassive heat removal systemHeat pipeNatural convectionNumerical simulation

I. Introduction

The International Atomic Energy Agency (IAEA) Nuclear Energy Safety Report^[1] highlights that in the future, more reliable, safer, and more efficient passive systems must be adopted for reactor designs to ensure that the decay heat of the reactor can be removed efficiently and passively under accident conditions^[2,3]. In recent years, heat pipes, invented at the Los Alamos National Laboratory in the 1960s, have been favored by nuclear scientists for efficient heat transfer. Their operation relies on gas–liquid two-phase flow of an internal working fluid, which transfers heat from one end of the heat pipe (evaporation section) to the other end (condensation section). Research has shown that the heat transfer capacity of heat pipes is much higher than that of metal materials. Initially, heat pipes were mainly used to solve heat dissipation problems of electronic equipment in spacecraft. With the advancement of heat pipe technology and the rapid development of the nuclear energy industry, the scope of heat pipe applications has expanded from traditional industrial sectors such as manufacturing, electronics, and energy to the design and application of large nuclear reactor systems such as space reactor optimization^[4,5], reactor containment^[6,7], small reactor core cooling^[8,9], and decommissioning after nuclear reactor accidents (e.g., Fukushima).

Preliminary neutronics and thermal hydraulic analysis shows that the temperature of fuel salt discharge during a molten salt reactor accident^[10,11] is 700–900 °C, while the effective temperature range of high temperature heat pipes is 400–2300 °C, which completely cover the temperature range of the fuel salt^[12,13]. In addition, high temperature heat pipes have many other advantages such as high thermal conductivity and ideal isothermality. The heat transfer mechanism of heat pipes relies on gravity or capillary pressure, which is ideal for passive operation. Finally, there is a temperature threshold for the operation of a heat pipe, which will start to work only when the temperature exceeds the threshold temperature, permitting effective temperature management. Based on these characteristics, the use of high temperature heat pipes for removing waste heat from molten salt reactors has significant advantages ^[14].

A conceptual design of a heat pipe passive residual heat removal system (PRHRS) for molten salt reactors has been proposed by Wang et al.^[15] and is shown in Fig. 1. The heat pipe passive waste heat removal system is mainly composed of a fuel salt drain tank, heat sink, and high temperature heat pipe. The drain tank is connected to the core and the first circuit of the molten salt reactor through a freeze valve. One end of the high temperature heat pipe is inserted into the drain tank, with the other end extending out of the tank. The drain tank and high temperature heat pipe are placed in the exhaust air duct. When the reactor temperature rises above a safe limit, which may result from damage to the primary loop, loss of flow accidents, or accidental fusing of freeze valves, the temperature of the fuel salt in the pressure vessel rises rapidly, which causes the freeze valve to melt and open rapidly. The fuel salt is then gravitationally discharged into a drain tank, and the high temperature heat pipes inserted into the drain tank begin to discharge waste heat from the accident into the atmosphere. The operation of the heat pipe type passive waste heat removal system is accomplished by three types of natural circulation (convection) couplings: 1) natural convection heat transfer between fuel salt and high temperature heat pipes in the drain tank; 2) gas–liquid two-phase natural circulation of the working fluid in the high temperature heat pipe; 3) natural circulation of air in the cooling air duct. The operation of the entire passive waste heat removal system does not require any external energy input, but only depends on the natural circulation (convection) of the system. Furthermore, compared to previous designs, the system structure has been greatly simplified, effectively avoiding failure of intermediate links, and is beneficial to future modular miniaturization designs of molten salt reactors.

Figure 1

Schematic diagram of heat pipe passive waste heat removal system for molten salt reactors.

However, numerical and experimental investigations concerning the natural convection heat transfer characteristics of fluoride have rarely been conducted, which has seriously hindered progress in the design of heat pipe passive residual heat removal systems for molten salt reactors. In this paper, a numerical study of heat transfer characteristics of a molten salt reactor is presented, which may provide the basis for the design of passive waste heat removal systems for molten salt reactors.

The paper is arranged as follows: Section 1 introduces the study and provides background information about the application of the system; numerical methodology and models are presented in Section 2; validation and verification of the numerical models are given in Sect. 3; in Sect. 4, the natural convection characteristics of a heat pipe-cooled residual heat removal system are obtained and discussed; finally, conclusions are presented in Sect. 5.

II. Numerical model and methodology

A. Experimental setup

An experimental system for validation of the heat pipe PRHRS was constructed and operated at Xi’an Jiaotong University (XJTU), China^[16]. The system consists of an argon gas loop and a molten salt loop. The latter includes two drain tanks, two freeze pipes, and a storage tank. A schematic representation of the drain tank with inserted heat pipes is shown in Fig. 2. To simulate the fission power produced by molten salt, nine electric heating rods are inserted into the drain tank. Eight heating rods are arranged symmetrically in a circle with radius of 0.18 m, while one heating rod is located in the center. The main structural parameters are listed in Table 1.

Figure 2

Schematic representation of the drain tank.

Model dimensions

Parameter	Dimension (mm)
Total height	920
Length of middle section	600
Inner radius	600
Wall thickness	10
Length of each heating rod	495

Six drain tanks were designed and simulated in the study, which are described as follows:

1) Single heat pipe at a height of 70 mm (Heat pipe A, abbreviated in Case 1).

2) Single heat pipe at a height of 200 mm (Heat pipe B, abbreviated in Case 2).

3) Single heat pipe at a height of 330 mm (Heat pipe C, abbreviated in Case 3).

4) With heat pipes A and C (spacing between heat pipes is 260 mm, abbreviated in Case 4).

5) With heat pipes A, B, and C (Spacing between heat pipes is 130 mm, abbreviated in Case 5).

B. Governing equations

The continuity equation, momentum equation, and energy equation are solved using the commercial CFD software ANSYS Fluent 15.0.

Continuity equation:

\frac{\partial ρ}{\partial τ} + \nabla \cdot (ρ \vec{v}) = 0,

(1)

where ρ and $\vec{v}$ represent density and velocity, respectively.

Momentum equations:

\frac{\partial \vec{u}}{\partial τ} + (\vec{v} \cdot \nabla) \vec{v} = \vec{f} - \frac{1}{ρ} \nabla p + \frac{μ}{ρ} \nabla^{2} \vec{v},

(2)

where μ represents viscosity, $\vec{f}$ represents external body forces, and ρ represents density.

Energy equation:

ρ c_{p} (\frac{\partial T}{\partial τ} + (\vec{v} \cdot \nabla) T) = λ \nabla^{2} T + \dot{ϕ},

(3)

where T and $\dot{ϕ}$ represent temperature and a dissipation function, respectively.

To address problems related to natural convection, the Boussinesq model is a good approximation to achieve faster convergence. In this model, density is kept constant in all solved equations, except a buoyancy term used in the momentum equation. The relationship is given as follows:

(ρ - ρ_{0}) ≫ - ρ_{0} β g \cdot Δ T,

(4)

where $ρ_{0}$ is the constant density, $β$ represents the coefficient of thermal expansion, g is the gravitational acceleration, and $Δ T$ represents the temperature difference. The boundary conditions and thermophysical properties of the fluoride salt (FLiNaK) are listed in Table 2 and Table 3.

Boundary conditions

Parameter	Range
Salt	FLiNaK (LiF-NaF-KF: 46.5-11.5-42 mol %)
Operating temperature (K)	743–923
Operating pressure (MPa)	0.12–0.2
Tank material	Type 316 stainless steel
Fluid level (m)	0.42
Power of heating rods (W)	2500–4500
Temperature range of heat pipes (K)	743–833

FLiNaK thermophysical properties

Parameter	Value
Melting point (℃)	454
Boiling point (℃)	1570
Density ρ (kg·m^-3)	2729.3-0.73·T
Viscosity μ (Pa·s)	2.487×10^-5exp(4478.62/T)
Thermal conductivity λ (W·m^-1·K^-1)	0.36+5.6×10^-4·T
Specific heat capacity c_p (J·kg^-1·K^-1)	976.78+1.0634·T

The pressure-based solver was used in this simulation, which is recommended for incompressible flows and a least square cell-based method, which has a faster convergence rate and is recommended for the Boussinesq model, was used for spatial discretization. In addition, the second-order upwind scheme was selected to solve the governing equations.

C. Grid independence analysis

Four sets of meshes were adopted to obtain grid-independent results. Results become grid independent at 2×10⁶ number of elements and any further increase of the number of nodes and elements only increased computational cost and time. There is no effect on the outlet temperature and the results become grid independent (Fig. 3a, b).

Figure 3

Grid independence study. (a) Case 4; (b) Case 5.

D. Data Processing

The natural convection heat transfer coefficient between fluoride salts and high temperature heat pipes can be calculated according to the following formulas:

\bar{N u} = \frac{h L}{λ} = \frac{Q}{A \bar{Δ T}} \times \frac{L}{λ},

(5)

Δ T = T_{s} - T_{w},

(6)

Q = q S,

(7)

S = 2 π r l,

(8)

where T_s is the fluoride salt temperature, h represents the heat transfer coefficient, T_w is the wall temperature, Q is the heat dissipation power of the side wall, S is the area of the side wall, q represents the heat flux of side wall, r is the inner radius of the closed chamber, and L represents the characteristic length of the heat pipe.

The ratio between viscosity and buoyancy forces is represented by the dimensionless Grashof number:

G r = g L^{3} α_{V} Δ T / v^{2},

(9)

where, g represents the gravitational acceleration, L is the characteristics length of the heat pipe, $α_{V}$ denotes the volumetric expansion coefficient, $Δ T$ is the temperature difference (see Eq. (6)), and ν represents the kinematic viscosity.

To measure the diffusion capacity of momentum and heat, the Prandtl number is used:

P r = μ c_{p} / λ,

(10)

where c_p represents the specific heat capacity and λ denotes the thermal conductivity of the working fluid. The Raleigh number is calculated using the Grashof and Prandtl numbers:

R a = G r \cdot P r

(11)

III. Model validation

To ensure reliability and accuracy of the numerical results, it is necessary to draw a comparison between experimental data and simulation results. Experimental data for single heat pipe cases (Case 1 through Case 3) were used. A comparison of static temperatures calculated through simulations with experimental data^[16] was conducted and the relative errors are shown in Table 4. Relative errors are calculated as follows:

Comparison between experimental data and calculation results

H (mm)	Case 1			Case 2			Case 3
	Exp. (K)	Cal. (K)	e (%)	Exp. (K)	Cal. (K)	e (%)	Exp. (K)	Cal. (K)	e (%)
415	879.25	883.32	0.46	874.28	880.23	0.68	876.29	882.12	0.67
285	871.66	869.25	-0.28	865.85	863.15	-0.31	865.68	866.94	0.15
155	861.09	858.74	-0.27	853.87	858.25	0.51	857.16	863.11	0.69
25	845.92	848.64	0.32	842.82	837.15	-0.67	847.77	843.85	-0.46
-40	842.88	845.15	0.27	840.92	838.15	-0.33	845.33	841.25	-0.48

e = | C a l . - E x p . | / E x p . \times 100 % .

(12)

The calculation results are in good agreement with the experimental data, demonstrating the accuracy of the simulation methodology.

IV. Natural convection characteristics analysis of the drain tank

In this section, characteristics of natural convection heat transfer between liquid fluoride (FLiNaK) and a vertical array of heat pipes with different heights in the drain tank are analyzed and discussed.

A. Parameter sensitivity analysis

Figure 4 (a) shows the axial temperature distribution of Case 4 at different fluoride salt temperatures and heating rod power settings. It can be seen that the axial temperature of fluoride salt gradually increases from low to high with increasing axial height, which is due to the natural circulation of fluoride in the drain tank. Figure 4(b) compares the temperature distribution of fluoride salts for cases where tube spacings are different while the salt bulk temperature is kept constant. When the overall average temperature of fluoride salt in the drain tank is the same, the temperature distribution difference of fluoride salt in the tank is very small, even when the spacing of vertical tube bundles in the drain tank is different. It should be noted that the maximum temperature difference between local average temperatures of fluoride salt using two different tube spacings (0.26 m and 0.13 m) is not more than 6 K.

Figure 4

Temperature distribution characteristics. (a) Molten salt temperature variations with axial height; (b) Variation of salt temperatures with different heat pipe spacing.

Figure 4 demonstrates that the axial temperature of fluoride salt varies uniformly with axial height in the straight section of the drain tank. The dimensionless temperature distribution of fluoride axial temperature and corresponding height in the straight barrel section of the drain tank is shown in Figure 5(a). The normalized temperature of fluoride salt increases gradually with an increase in normalized height. Similar to the results for natural convection heat transfer between fluoride salt and a single heat pipe, Figure 5(a) shows that regardless of the spacing of the heat pipe, the overall average temperature of fluoride salt and the power of the heating rods are different. The dimensionless distribution of the axial temperature of fluoride salt at different working conditions increases steadily from low to high, and the dimensionless average temperature of fluoride salt in the straight section of drain tank shows the following linear relationship:

Figure 5

Temperature difference analysis. (a) Dimensionless axial temperature distribution of fluoride salt; (b) Variations of the temperature difference of fluoride salt relative to the average temperature of fluoride salt; (c) Variations of the temperature difference of fluoride salt relative to the average power setting of the heat pipes.

T_{s} / T_{b} = 0.04975 \cdot H / H_{\max} + 0.98011,

(13)

where the T_b is the bulk temperature of the molten salt.

Figure 5(b) shows the temperature difference variation of fluoride salt relative to the average temperature of fluoride salt inside the tank, which was measured with the help of measuring points in the drain tank. Figure 5(c) shows the temperature difference variation of fluoride salt relative to the average power setting of the heat pipes. It is evident that the temperature difference of fluoride salt remains essentially unchanged under all working conditions.

B. Analysis of heat transfer characteristics

On the basis of the simulation results, the natural convection process between fluoride salt and heat pipes was further analyzed. The natural convection heat transfer coefficient between liquid fluoride (FLiNaK) and two differently spaced vertical heat pipe bundles (Case 4 and Case 5) is shown in Figure 6(a). Natural convection heat transfer coefficients for fluoride salt and the vertical tube bundles remain similar for both cases, with no obvious difference between heat transfer coefficients for different heat pipe spacings.

Figure 6

Heat transfer characteristics. (a) Comparison of heat transfer coefficients for a single heat pipe versus vertical arrays of heat pipes; (b) Average heat dissipation power (P) of vertical arrays of heat pipes compared with a single heat pipe; (c) Relationship between Nu and Ra numbers.

With an increase in fluoride salt temperature, natural convection becomes increasingly intense, leading to increases in the natural convection heat transfer coefficient and convection intensity. When the overall average temperature of fluoride salt in drain tank increases from 790 K to 850 K, the resulting value for the natural convection heat transfer coefficient increases from 620 W·m^-2·K^-1 to 710 W·m^-2·K^-1. In Figure 6(a), values for the natural convection heat transfer coefficient of a single heat pipe are shown for comparison with the vertical heat pipe arrays. For a single heat pipe, the overall average temperature of fluoride salt increases from 779 K to 880 K and as a result the natural convection coefficient increases from 720 W·m^-2·K^-1 to 790 W·m^-2·K^-1. It is clear that the natural convection heat transfer coefficient of fluoride salt using vertical tube bundles is significantly (15 %) lower compared to a single heat pipe setup.

Figure 6(b) shows the variation of the average heat dissipation power of heat pipes relative to the temperature difference between the fluoride salt and the average wall temperature of the heat pipe under different conditions. It is evident that the average heat dissipation power between fluoride salt and heat pipes is lower for the vertical arrays of heat pipes compared with that of a single heat pipe. Although the average power of the heating rods for the vertical arrays of heat pipes is higher, the heat dissipation power for a single heat pipe is 100 W higher than that for the vertical arrays of heat pipes. The decrease in average heat dissipation power and convective heat transfer coefficient for the vertical arrays of heat pipes indicates that the natural convection intensity of fluoride salt is weakened as shown in Figure 6(b). There may be two reasons for this behavior: (1) the arrangement of the vertical bundles of heat pipes interferes with the natural convection flow of the fluoride salt, resulting in a decrease of the average convection heat transfer capacity between fluoride salt and each heat pipe. Similar experiments with water as working fluid show that the wake generated by water flowing through the lower tube affects the flow of water outside the upper tube, hindering heat transfer and reducing heat dissipation power^[17]; (2) flow constriction limits the heat transfer intensity. In a closed and narrow drain tank, natural convection between the tube bundle and fluoride salt cannot fully develop, limiting flow and heat transfer^[18].

Figure 6 (c) shows average Nusselt numbers (Nu) representing natural convection heat transfer between fluoride salt and vertical tube bundles relative to Rayleigh numbers for different tube spacings. It is evident that Nu increases with increasing Ra numbers. Due to a weakening of the overall natural convection intensity, average Nusselt numbers vary from 23 to 25, which is significantly smaller than for a single heat pipe.

V. Conclusion

Numerical investigations of natural convection heat transfer characteristics of fluoride salt in a drain tank were conducted in this study. The simulation methodology was validated by comparing the calculation results with experimental data. Main conclusions are summarized as follows:

1. The radial temperature of liquid fluoride salt in the drain tank shows a uniform distribution; however, the salt temperature increases with increasing axial height from bottom to top.

2. The average temperature of fluoride salt and the average heat dissipation power of the heat pipe have no obvious influence on temperature differences within the tank.

3. The natural convection heat transfer coefficient increases with increasing temperature of the fluoride salt. The natural convection intensity increases with an increase in the height of the heat pipe in the tank. The spacing between heat pipes shows no obvious effect on the natural convection heat transfer coefficient.

Based on these conclusions, several suggestions for the design of a PRHRS can be made: (1) the vertical position of heat pipes should be reasonably high to enhance natural convection heat transfer; and (2) to increase heat transfer, the number of heat pipes should be increased considering that heat pipe spacing has a negligible effect on heat transfer.

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