Keywords

1 Introduction

The heat pipe cooled reactor is characterized by a simple system, high reliability, passive heat transfer, and a high operating temperature. There is a wide range of potential application scenarios in the field of micro-nuclear power in extreme environments such as deep space, deep sea, and deep earth [1,2,3,4]. In heat pipe reactors, alkali metal heat pipes are the critical heat transfer elements. Heat is transferred efficiently through these heat transfer elements based on latent heat. Additionally, capillary force drives the internal medium into continuous phase transitions. Therefore, heat pipes have high heat transfer performance [5], isothermal performance [6], and thermal response speed [7]. Due to their excellent heat transfer capabilities, heat pipes have been widely used in waste heat recovery [8,9,10], nuclear reactor power [11,12,13,14], and residual heat removal [15, 16].

In general, heat pipes provide a high level of robustness and repeatability. Many steady-state and transient analyses have demonstrated the high level of safety and reliability of the heat pipe reactor. For example, Poston et al. [17] described the design and analysis of the SAFE-400 reactor. They analyzed heat pipe failure scenarios and pointed out that the SAFE-400 was designed with a considerable thermal margin under nominal operating conditions. Li et al.[18] developed a transient analysis code SNPS-FTASR to study the transient response of HP-STMCs core on reactivity-insertion accident transient and a heat pipe failed transient. The simulation showed that the HP-STMCs core was safe at transient states including reactivity-insertion accidents and a heat pipe failure. Ma et al. [12] also conducted the transient heat pipe failure accident analysis of the MegaPower reactor. Their results showed that the operating conditions far from the failure area were little affected.

However, heat pipes can experience a particular unstable phenomenon, known as geyser boiling, which leads to temperature oscillations in heat pipes. Murphy [19] described the geyser boiling phenomenon as the rapid expulsion of a boiling liquid and its vapor from a vertical heat pipe and mentioned that geyser boiling behavior can be affected by many factors. Lin et al. [20] experimentally studied the influence of heating power, condenser temperature, liquid charge ratio, and evaporator length on water and ethanol geyser boiling. Ma et al. [21] mainly considered the geyser boiling in sodium heat pipes at near-horizontal inclination angles. They found that the evaporation pulsation was up to 80 ℃ in the operating temperature range of 600 ℃–700 ℃.

In a heat pipe cooled reactor, alkali metal heat pipes are closely related to the reactor operation. During geyser boiling, the heat pipe's temperature oscillates, which inevitably affects the core's operation status. However, previous studies of reactor safety have barely considered the instability of heat pipe temperature oscillations. Meanwhile, in reactor design, safety is an absolute priority and extremely strict. Therefore, further investigations on the effects of heat pipe temperature oscillations are essential.

In the present work, a coupled neutronic and thermal-mechanical model was developed to study the effects of geyser boiling and temperature oscillations on the thermal-mechanical performance of the reactor core. The sensitivity analysis of the oscillation period was also conducted.

2 Methodology

Figure 1 shows the model relationship, developed in this work, which mainly includes the heat pipe heat transfer model, neutronic point kinetics model, neutron transport model, and core zonal multi-channel thermal-mechanical model. The neutronic point kinetics model determines the total power with the neutron transport simulation to obtain the relative power distribution. The power is transferred as a volumetric heat source to the core zonal multi-channel thermal-mechanical model to determine the temperature field and force field of the heat pipe, fuel, monolith, and other components, and fed back to the neutronic point kinetics model via reactivity feedback.

These models are discussed in detail in the following sections.

Fig. 1.
figure 1

Coupled neutronic and thermal-mechanical model

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2.1 Point Kinetics Model

A point kinetics model is used to determine the fission power. The standard point kinetics model is used with six groups of delayed neutrons to calculate the core fission power as:

$$ \begin{aligned} & \frac{{{\text{d}}N(t)}}{{{\text{d}}t}} = \frac{{\rho (t) - \overline{\beta }(t)}}{\Lambda (t)}N(t) + \sum\limits_{i = 1}^{d} {\lambda_{{\text{i}}} } C_{{\text{i}}} (t) \\ & \frac{{\partial C_{{\text{i}}} (t)}}{\partial t} = \frac{{\overline{\beta }_{{\text{i}}} (t)}}{\Lambda (t)}N(t) - \lambda_{{\text{i}}} C_{{\text{i}}} (t) \\ \end{aligned} $$
(1)

where \(N(t)\) is the approximate neutron density of each energy group, \(t\) is the time, \(\rho\) is the reactivity, \(\beta_{i}\) is the fraction of effective delayed neutrons, \(\Lambda\) is the neutron generation time, \(\lambda_{i}\) is the decay constant of the delayed neutrons, and \(C_{i}\) is the delayed neutron precursor concentration. The Monte Carlo program RMC is used to determine the dynamics parameters in Eq. (1).

2.2 Reactivity Feedback Model

The Doppler effect and thermal expansion are the primary mechanisms for reactivity feedback. Besides, in the heat pipe cooled reactor, the heat pipe two-phase working fluid is impacted by the operating temperature, introducing a reactivity temperature coefficient. Therefore, the reactivity feedback equation can be written as:

$$ \begin{aligned} \rho (t) & = \rho_{{{\text{ext}}}} + \rho_{{{\text{feedback}}}} \\ & = \rho_{{{\text{ext}}}} + \alpha_{{{\text{f}},{\text{D}}}} \ln \left( {\overline{{T_{{\text{f}}} }} /T_{{\text{f}}}^{0} } \right) + \alpha_{{{\text{f}},{\text{G}}}} \left( {\overline{{T_{{\text{f}}} }} - T_{{\text{f}}}^{0} } \right) \\ & \quad + \alpha_{{{\text{M}},{\text{D}}}} \ln \left( {\overline{{T_{{\text{M}}} }} /T_{{\text{M}}}^{0} } \right) + \alpha_{{{\text{M}},{\text{G}}}} \left( {\overline{{T_{{\text{M}}} }} - T_{{\text{M}}}^{0} } \right) \\ & \quad + \alpha_{{{\text{R}},{\text{D}}}} \ln \left( {\overline{{T_{{\text{R}}} }} /T_{{\text{R}}}^{0} } \right) + \alpha_{{{\text{R}},{\text{G}}}} \left( {\overline{{T_{{\text{R}}} }} - T_{{\text{R}}}^{0} } \right) \\ & \quad + \alpha_{{{\text{HP}}}} \left( {\overline{{T_{{{\text{HP}}}} }} - T_{{{\text{HP}}}}^{0} } \right) \\ \end{aligned} $$
(2)

where \(\overline{T}\) and \(T^{0}\) are the average temperature and the reference temperature. Subscripts f, M, R and HP refer to the fuel, the monolith, the reflector, and the heat pipe, respectively. Subscripts D and G refer to the Doppler reactivity coefficient and geometric reactivity coefficient.

2.3 Zonal Multi-channel Thermal-Mechanical Model

Temperature changes can lead to changes in geometric parameters such as radius and air gap thickness and material properties such as density. These changes, in turn, affect the temperature distribution in the core. Therefore, thermal-mechanical coupled analyses are essential for heat pipe cooled reactors. To simplify this coupling problem, the core is divided into several zones, and each zone is further divided into several channels. Each channel's thermal and mechanical models are constructed and then coupled.

2.3.1 Heat Transfer Model in a Single Channel

A hexagonal single heat transfer unit, as illustrated in Fig. 2, can be extracted from a typical heat pipe cooled reactor and contains a heat pipe, its adjacent monolith, and fuel rods. Each unit is then modeled as a single channel based on volume equivalence. Figure 2 depicts the modeled single channel with control volumes separated axially. The heat pipe is in the middle with layers for the heat pipe wall, monolith, helium gap, and fuel. The heat conduction equation in the radial direction in cylindrical coordinates is:

$$ \rho_{{\text{i}}} c_{{\text{i}}} \frac{{\partial T_{{\text{i}}} }}{\partial t} = \frac{1}{r}\frac{\partial }{\partial r}\left( {\lambda_{{\text{i}}} r\frac{{\partial T_{{\text{i}}} }}{\partial r}} \right) + \dot{\phi }_{{\text{i}}} $$
(3)

where the subscript \(i\) represents the various structures such as the monolith, helium gas gap, fuel, or heat pipe wall; \(\rho\) is the density; \(c\) is the specific heat; \(\lambda\) is the thermal conductivity; \(T\) is the material temperature, and \(\dot{\phi }\) is the power density including the fission heat source and the radial thermal conduction between components.

The boundary conditions have continuity of the temperature and heat flux at the contact surfaces of the walls. The outer channel wall, which corresponds to the fuel pellet center, is assumed to be adiabatic due to symmetry.

Fig. 2.
figure 2

A fuel assembly and its single-channel model

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This modeling changes the geometry of the air gap and the fuel, which affects the thermal resistance and temperature distribution. Therefore, according to the conservation conditions of heat flow and the principle of equivalence with thermal resistance, a thermal conductivity correction is introduced to correct the heat transfer results:

$$ \left\{ {\begin{array}{*{20}l} {\lambda_{{\text{f}}}^{\prime } = k_{{\text{f}}} \lambda_{{\text{f}}} = \frac{{\frac{{\dot{\phi }}}{4}\left( {r_{{{\text{f}},{\text{o}}}}^{2} - r_{{{\text{f}},{\text{i}}}}^{2} } \right) - \frac{{\dot{\phi }}}{2}r_{{{\text{f}},{\text{i}}}}^{2} \ln \left( {\frac{{r_{{{\text{f}},{\text{o}}}} }}{{r_{{{\text{f}},{\text{i}}}} }}} \right)}}{{\frac{{\dot{\phi }}}{{4\lambda_{{\text{f}}} }}r_{{\text{f}}}^{2} }} = \frac{{\left( {r_{{{\text{f}},{\text{o}}}}^{2} - r_{{{\text{f}},{\text{i}}}}^{2} } \right) - 2r_{{{\text{f}},{\text{o}}}}^{2} \ln \left( {\frac{{r_{{{\text{f}},{\text{o}}}} }}{{r_{{{\text{f}},{\text{i}}}} }}} \right)}}{{r_{{\text{f}}}^{2} }}\lambda_{{\text{f}}} } \hfill \\ {\lambda_{{\text{g}}}^{\prime } = k_{{\text{g}}} \lambda_{{\text{g}}} = \frac{{6\ln \left( {r_{{{\text{g}},{\text{o}}}} /r_{{{\text{g}},{\text{i}}}} } \right)/(2\pi L)}}{{3\ln \left( {\left( {r_{{\text{f}}} + 2r_{{\text{g}}} } \right)/r_{{\text{f}}} } \right)/\left( {2\pi L\lambda_{{\text{g}}} } \right)}} = \frac{{2\ln \left( {r_{{{\text{g}},{\text{o}}}} /r_{{{\text{g}},{\text{i}}}} } \right)}}{{\ln \left( {\left( {r_{{\text{f}}} + 2r_{{\text{g}}} } \right)/r_{{\text{f}}} } \right)}}\lambda_{{\text{g}}} } \hfill \\ \end{array} } \right. $$
(4)

where \(\lambda\) is the thermal conductivity, and \(k\) is the correction factor. Subscripts f and g refer to the fuel and gas gap. \(L\) is the axial height. \(r\) is the radius. Subscripts o and i refer to the outer and inner radius, respectively.

2.3.2 Mechanical Model in a Single Channel

The mechanical model unit is extracted according to the repeating geometric layout of the heat pipe reactor, which differs from the heat transfer model in that the fuel is located in the center and the monolith surrounds it, as shown in Fig. 3.

Fig. 3.
figure 3

Mechanical model

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Basic assumptions such as generalized plane strain assumption, continuity assumption, and quasi-static assumption are introduced [22], then the balance equation, geometric equation, and constitutive equation are as follows:

$$ \frac{{\partial \sigma_{r} }}{\partial r} + \frac{{\sigma_{r} - \sigma_{\theta } }}{r} = 0 $$
(5)
$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{r} = \frac{\partial u}{{\partial r}}} \hfill \\ {\varepsilon_{\theta } = \frac{u}{r}} \hfill \\ {\varepsilon_{z} = \frac{\partial w}{{\partial z}} = {\text{const}}} \hfill \\ \end{array} } \right. $$
(6)
$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{{\text{r}}} = \varepsilon_{r}^{e} + \varepsilon_{r}^{\alpha } + \varepsilon_{r}^{ie} = \frac{1}{E}\left( {\sigma_{r} - \mu \left( {\sigma_{\theta } + \sigma_{z} } \right)} \right) + \varepsilon_{r}^{\alpha } + \varepsilon_{r}^{ie} } \hfill \\ {\varepsilon_{\theta } = \varepsilon_{\theta }^{e} + \varepsilon_{\theta }^{\alpha } + \varepsilon_{\theta }^{ie} = \frac{1}{E}\left( {\sigma_{\theta } - \mu \left( {\sigma_{r} + \sigma_{z} } \right)} \right) + \varepsilon_{\theta }^{\alpha } + \varepsilon_{\theta }^{ie} } \hfill \\ {\varepsilon_{z} = \varepsilon_{z}^{e} + \varepsilon_{x}^{\alpha } + \varepsilon_{z}^{ie} = \frac{1}{E}\left( {\sigma_{z} - \mu \left( {\sigma_{r} + \sigma_{\theta } } \right)} \right) + \varepsilon_{z}^{\alpha } + \varepsilon_{z}^{ie} } \hfill \\ \end{array} } \right. $$
(7)

where \(\sigma\) is principal stress and \(\varepsilon\) represents the strain. Subscripts \(r\), \(\theta\) and \(z\) refer to the radial, circumferential, and axial terms, respectively. \(\varepsilon^{e}\), \(\varepsilon^{\alpha }\) and \(\varepsilon^{ie}\) are the elastic strain, thermal expansion strain, and other inelastic strains. \(u\) and \(w\) are the radial and axial displacements, respectively; \(\mu\) is Poisson’s ratio, and \(E\) is the elastic modulus.

The boundary conditions of the fuel pellet center, the outer side of the fuel, and the inner side of the monolith are as follows:

$$ \begin{aligned} & \sigma_{{r,\,{\text{in}}}}^{{{\text{pellet}}}} = \sigma_{{\theta ,{\text{ in}}}}^{{{\text{pellet}}}} \\ & \sigma_{{r,{\text{ in }}}}^{{{\text{monolith}}}} = \sigma_{{r,{\text{ out }}}}^{{{\text{pellet}}}} = \left\{ {\begin{array}{*{20}l} { - P_{{\text{g}}} ,d_{{\text{g}}} > 0} \hfill \\ { - P_{{\text{con }}} ,d_{{\text{g}}} = 0} \hfill \\ \end{array} } \right. \\ \end{aligned} $$
(8)

where \(d_{{\text{g}}}\) is the air gap thickness. When the fuel and the monolith contact with each other, \(d_{{\text{g}}} = 0\) and the radial stress is the contact stress of the fuel and the monolith(\(P_{{{\text{con}}}}\)). Otherwise, \(d_{{\text{g}}} > 0\) and the radial stress is the gas pressure (\(P_{{\text{g}}}\)).

2.3.3 Thermal and Mechanical Coupling

The thermal and mechanical results for the reactor are then coupled via variable transfer, as illustrated in Fig. 4. The core temperature field calculated by the heat transfer model is transferred to the mechanical model as a thermal load, and the core stress and strain calculated by the mechanical model will update the fuel radius, air gap thickness, and material density in the heat transfer model.

Fig. 4.
figure 4

Relationship between the thermal and mechanical analyses

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2.3.4 Core Zonal Multi-channel Thermal-Mechanical Model

The core zonal multi-channel thermal-mechanical model is used to reflect the radial heat transfer between channels. In a typical heat pipe reactor, the fuel rods and heat pipes are organized within the monolith in a regular, compact way that may be separated into hexagonal zones, as shown in Fig. 5a. Figure 5b illustrates the extraction of a single channel for transient thermal analysis within each hexagonal zone. The operational statuses of all channels within a zone are assumed to be identical due to symmetry. The heat transfer between channels inside a zone is thus not taken into account; only the heat transfer between zones is.

The different zones are modeled using the equal volume method as:

$$ SN_{i} = \pi \left( {r_{i}^{2} - r_{i - 1}^{2} } \right) $$
(9)

where \(S\) is the area of one fuel unit, and \(N\) is the number of units contained in the zone. \(r\) is the outer wall radius corresponding to the zone, and \(i\) represents different zones.

The total heat source is the sum of the fuel fission power inside the zone and the conduction heat with adjacent zones:

$$ \phi_{i,j} = \phi_{i,j}^{{\text{fuel }}} + \phi_{i,j}^{{{\text{con}}}} = \phi_{i,j}^{{\text{fuel }}} + \sum\limits_{k = 1}^{m} {\lambda_{k,i} } A_{k,i} \frac{{\overline{{T_{k,j} }} - \overline{{T_{i,j} }} }}{{\Delta \overline{{r_{i,k} }} }} $$
(10)

where \(\phi\) is net heat flux, \(T\) is volume average temperature, and \(A\) is the heat transfer area. \(\lambda\) is the conductivity, which is temperature-dependent and determined by the volume equivalence. Subscripts \(i\) and \(j\) refer to the zone number and the axial position. Subscript \(k\) represents the zone around the zone \(i\).

Fig. 5.
figure 5

Multi-channel heat transfer model

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The heat source term of a unit is:

$$ \phi_{{i,j,{\text{single}}}} = \phi_{i,j} /N_{i} $$
(11)

where \(\phi_{{i,j,{\text{single}}}}\) represents the power of axial control volume j in the single channel of the zone \(i\), and \(N_{i}\) is the number of single channels in the zone \(i\).

Through the helium gap, heat is transferred from the side channel into the reflector and then to the environment via convection.

3 Simulations and Analyses

The MegaPower reactor is a heat pipe cooled reactor designed by Los Alamos National Laboratory to meet off-grid areas’ specialized energy demands [23]. Numerous calculations have been published on the neutron physics and thermal engineering of the MegaPower reactor [23, 24]. The above models and methods will be applied to the MegaPower to analyze the effects of heat pipe temperature oscillations on the reactor core.

As shown in Fig. 6, the core is divided into 16 zones with zone #1 as the hot channel, zone #2 to zone #15 as the transition channels, and zone #16 at the edge. The reactor heat is removed by solid heat conduction to the environment through the outer reflector wall by natural air convection to the ambient temperature at 300 K with a convection heat transfer coefficient of 20 \({\text{W}}/\left( {{\text{m}}^{2} \cdot {\text{k}}} \right)\).

3.1 Heat Pipe Temperature Oscillation

Figure 7 illustrates the heat pipe temperature oscillation behavior. Table 1 lists the temperature oscillation parameters for the case of a 30° inclination angle [21]. The temperature oscillation in this work is described using a power function as follows:

$$ \begin{aligned} & A_{1} (h) = \left\{ {\begin{array}{*{20}l} {A_{1,\max } \frac{h}{H}\left( {\frac{t}{\alpha T}} \right)^{\beta } ,t < \alpha T} \hfill \\ {A_{1,\max } \frac{h}{H}\left( {\frac{T - t}{{(1 - \alpha )T}}} \right)^{\gamma } ,t > \alpha T} \hfill \\ \end{array} } \right. \\ & A_{{\text{v}}} = \left\{ {\begin{array}{*{20}l} {A_{{{\text{v}},\max }} \left( {\frac{t}{\alpha T}} \right)^{\beta } ,t < \alpha T} \hfill \\ {A_{{{\text{v}},\max }} \left( {\frac{T - t}{{(1 - \alpha )T}}} \right)^{\gamma } ,t > \alpha T} \hfill \\ \end{array} } \right. \\ \end{aligned} $$
(12)

where \(A_{1} (h)\) is the temperature amplitude at height \(h\) in the liquid pool, \(H\) is the maximum height of the liquid pool. \(A_{{\text{v}}}\) is the temperature amplitude in the vapor area. \(A_{1,\max }\) and \(A_{{{\text{v}},\max }}\) represent the corresponding maximum temperature amplitudes, respectively. \(\alpha\) represents the ratio of rise time in a single cycle. \(\beta\) and \(\gamma\) are power function coefficients of 0.5 and 1, respectively, determined by the experimental results. \(T\) is the period of geyser boiling, and \(t \in [0,T]\).

Fig. 6.
figure 6

Core zones partitioning

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Table 1. Oscillation parameters for a 30° inclination angle
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Fig. 7.
figure 7

Temperature oscillation of heat pipes

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Fig. 8.
figure 8

The effect of temperature oscillation on the core

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3.2 Effect of Heat Pipe Temperature Oscillation on the Core

The heat pipe reactor uses the heat pipe as the heat transmission element. The changes in heat pipe temperature will inevitably cause the core temperature and the core power to fluctuate. Besides, the changes will also affect the stress in the core. Therefore, this paper simulates the thermal and mechanical effects of heat pipe temperature oscillations on the core separately. The simulation parameters are determined by the above experimental results with an oscillation period of 60 s, a liquid pool height of 1/4 of the evaporator, and vapor area and liquid pool amplitudes of 10 ℃ and 70 ℃, respectively. Finally, a sensitivity analysis of the period in the oscillation parameters is performed.

3.2.1 Thermal Effect

Figure 8 shows the effect of heat pipe temperature oscillation on the core power and reactivity. At 100 s, the geyser boiling starts, which introduce a small positive reactivity and thus increase the core power. However, negative feedback, such as the Doppler effect, causes the core to return to the critical gradually. Then, the heat pipe temperature oscillation still affects the core, resulting in a steady fluctuation in core power with a period of 60 s and an amplitude of 25 kW.

Figure 9a shows the temperature fluctuation of the heat pipe’s outer wall. As height rises, the amplitude decreases. The temperature amplitude is 60 ℃ at a height of 0.15 m from the bottom of the evaporator and falls to 10 ℃ at a height of 0.375 m.

Figure 9b shows the temperature fluctuation of the monolith wall near the fuel. It can be seen that the effect of heat pipe temperature oscillation is slightly suppressed. The wall's temperature amplitude drops to 47 ℃ (at 0.15 m) and 9 ℃ (at 0.375 m or higher). Figure 9c and Fig. 9d show the temperature fluctuation of the fuel outer wall and the fuel pellet center, respectively. The influence is further attenuated. At a height of 0.15 m, the fuel outer wall temperature amplitude falls to 30 ℃ and the center temperature amplitude falls to 22 ℃.

Fig. 9.
figure 9

Core temperature oscillation for geyser boiling condition (zone #1)

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3.2.2 Mechanical Effect

Figure 10 shows the effect of heat pipe temperature oscillation on the stress in the core. With an amplitude of about 6 MPa, the peak stress oscillates at the same frequency as the temperature for both the fuel and the monolith. Although the oscillation range remains within the safety limit, the oscillation may aggravate the material thermal fatigue and cause structural failure during the long life of heat pipe reactors, which threatens the core safety. Therefore, temperature oscillations should be avoided in heat pipe reactors.

Fig. 10.
figure 10

Stress oscillation for geyser boiling condition (zone #1)

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3.2.3 Sensitivity Analysis of the Period

For sensitivity analysis, the oscillation period is adjusted to 1/4 of the original, that is, 15 s. The results are shown in Fig. 11. The temperature oscillation amplitude of the fuel and the monolith is significantly reduced as the period decreases. When the period is reduced from 60 s to 15 s, the temperature amplitude of the fuel pellet center at a height of 0.15 m decreases from 22 ℃ to 5 ℃. Similar trends are seen in the other areas. Therefore, long-period heat pipe temperature oscillations are more likely to transmit temperature fluctuations to the core, and high-frequency heat pipe temperature oscillations can be absorbed by the core heat capacity and their effects are mitigated.

Fig. 11.
figure 11

Temperature oscillation of monolith and fuel with a period of 15 s

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4 Conclusions

This work introduces a coupled neutronic and thermal-mechanical analysis method of heat pipe reactors. The megawatt heat pipe reactor, MegaPower, was chosen as an application case to study the influence of heat pipe temperature oscillations caused by geyser boiling on the critical parameters during reactor operation, such as temperature and stress. The calculations show that:

  1. (1)

    The heat pipe temperature oscillation has a thermal effect on the core, causing the monolith temperature and fuel temperature to oscillate at the same frequency. Besides, as the oscillation period decreases, the temperature amplitude also decreases due to the heat capacity of the core. The core temperature amplitudes are 22 ℃ and 5 ℃, respectively, for oscillation periods of 60 s and 15 s, which indicates that the effects will be mitigated for low-period heat pipe temperature oscillations.

  2. (2)

    Due to the temperature oscillation in the core, the peak stresses in the monolith and the fuel oscillate at the same frequency. The oscillation amplitude is about 6 MPa at a period of 60 s. Although the fluctuation range remains within the safety limit, this fluctuation may aggravate thermal fatigue, which may result in structural failure for a long period of operation. Therefore, the geyser boiling must be avoided during the steady-state operation of a heat pipe cooled reactor.