Keywords

1 Introduction

With the increasingly energy shortage and environmental problems, and the large-scale development of nuclear power is facing challenges such as safety, economy, nuclear waste treatment and nuclear proliferation prevention, the concept of traveling wave reactor [1] has been put forward internationally. Traveling wave reactor is a self-sustaining Breed-and-Burn (B&B) reactor. Its basic concept comes from proliferation combustion reactor. Its theoretical basis is that the reactor core starts from highly enriched fuel, the superfluous fast neutrons produced by the fission reaction convert the surrounding U-238 into Pu-239 (or Th-232 into U-233). When Pu-239 (or U-233) reaches a certain concentration, a new fission reaction is formed, that is, and the fission reaction generates superfluous fast neutrons again to realize the proliferation of nuclear fuel. In the process of proliferation and burning, the traveling wave is continuously moved along the axis throughout its life.

The coolant liquid metal lead used in lead cooled fast reactor has strong chemical inertness. Its good thermodynamic properties provide a guarantee for the full heat removal of the reactor core, which is conducive to the design of the reactor core with higher power density, and conforms to the characteristics of higher power density in the combustion zone of traveling wave reactor. At the same time, due to the small moderation and absorption of neutrons by lead, the high neutron economy in the core of lead cooled fast reactor is conducive to the long-life operation of traveling wave reactor.

In 2006, Sekimoto [2] carried out the research on the core design and safety characteristics of lead-bismuth cooled traveling wave reactor. The research shows that during the whole life of the traveling wave reactor, the neutron fluence rate or power in the core can maintain a stable shape to propagate very slowly, because of this characteristic of traveling wave reactor, Sekimoto calls axial traveling wave reactor as CANDLE. (Constant Axial shape of Neutral flux nuclide, number identities and power shape During Life of Energy producing reactor).

In order to better understand and master the thermal phenomena of lead cooled traveling wave reactor, it is necessary to develop a more accurate thermal hydraulic calculation model. Taking the lead cooled traveling wave reactor core as the object, according to the investigation and investigation of the thermal hydraulic model of the lead based fast reactor, this paper develops a program suitable for the thermal hydraulic calculation of the lead cooled traveling wave reactor core, applies the developed thermal program to carry out the thermal hydraulic analysis of the lead cooled traveling wave reactor, and studies and analyzes the safety and temperature field distribution characteristics of the lead cooled traveling wave reactor.

2 Thermal-Hydraulic Model

2.1 Model of Coolant

Due to the closed fuel assembly used in lead cooled traveling wave reactor, the exchange of coolant momentum and mass between the assemblies is not considered.

In this paper, based on the finite volume method, the mass, momentum and energy conservation equations of coolant are established, which are solved by axial propulsion in space and implicit scheme in time.

Mass conservation equation:

$$ \frac{{\partial m_{t} }}{\partial t} = \sum\limits_{i = 1}^{N} {\frac{{\partial m_{i} }}{\partial t}} $$
(1)

In the formula (1), \(m_{t}\) refers to the total mass flow of inlet coolant, \(m_{i}\) refers to the mass flow of coolant in the \(i\) channel, and \(N\) refers to the number of core channels.

Momentum conservation equation:

$$ \begin{gathered} \frac{1}{{A_{i} }}\frac{{\partial m_{i} }}{\partial t} + \frac{\partial }{\partial z}\left( {\frac{{m_{i}^{2} }}{{\rho A_{i}^{2} }}} \right) = - \frac{\partial p}{{\partial z}} - \rho g \\ - \frac{{fm_{i}^{2} }}{{2d_{e} \rho A_{i}^{2} }} \\ \end{gathered} $$
(2)

In the formula (2), \(A_{i}\) is the area of the \(i\) channel, \(\rho\) is the coolant density, \(p\) is the coolant pressure, \(f\) is the friction coefficient, and \(d_{e}\) is the equivalent diameter of the channel.

Energy conservation equation:

$$ \rho \frac{\partial h}{{\partial t}} + \frac{\partial }{\partial z}\left( {\frac{{w_{i} h}}{{A_{i} }}} \right) = q\left( t \right) $$
(3)

In the formula (3), \(q\left( t \right)\) is the heat absorbed by the coolant in the control body. The first item on the left represents the increment of heat in the control body over time, the second item represents the heat of coolant flowing in and out of the control body, and the right item represents the heat absorbed by the coolant.

The core calculation program HANDF [6] and the thermal hydraulic program developed in this paper are used for code-to-code verification, and the coolant temperature and fuel temperature of each component are compared. The temperature calculated by the program in this paper is in good agreement with the calculated value of HANDF, and the maximum relative deviation is less than 0.5%, which proves that the model of the program is reliable and stable, and can be used for the thermal hydraulic analysis and calculation of the reactor core.

2.2 Model of Fuel Rod

For the heat conduction calculation of fuel rod, the circumferential heat conduction is ignored, and only the radial and axial heat conduction of fuel are considered. For the air gap and cladding, the thermal resistance model is used. The governing equation of the two-dimensional heat conduction model is:

$$ \begin{gathered} \rho c_{p} \frac{\partial T}{{\partial t}} = \frac{1}{r}\frac{\partial }{\partial r}\left( {\lambda \cdot r \cdot \frac{\partial T}{{\partial r}}} \right) + \\ \frac{\partial }{\partial z}\left( {\lambda \cdot \frac{\partial T}{{\partial z}}} \right) + S \\ \end{gathered} $$
(4)

In the formula (4), \(\rho\) is the density of the fuel rod, \(c_{p}\) is the heat capacity of the fuel rod, \(T\) is the temperature of the fuel rod, \(\lambda\) is the thermal conductivity of the fuel rod, and \(S\) is the volumetric heat release rate.

2.3 Model of Flow Distribution

In the fuel assembly pressure drop calculation, the total pressure drop of the channel is composed of gravity pressure drop, friction pressure drops, acceleration pressure drop and local pressure drop. Set the boundary condition that the pressure of each channel at the inlet and outlet is the same. Control equation of flow dynamic distribution:

$$ \begin{gathered} \frac{L}{{A_{i} }}\frac{{\partial m_{i} }}{\partial t} = \left( {P_{in} - P_{out} } \right) - \int\limits_{0}^{L} {\frac{\partial }{\partial z}\frac{{m_{i}^{2} }}{{\rho A_{i}^{2} }}} dz - \\ \int_{0}^{L} {\frac{{fm_{i}^{2} }}{{2d_{e} \rho A_{i}^{2} }}dz} - \int\limits_{0}^{L} {\rho gdz} \\ \end{gathered} $$
(5)

In the formula (5), \(L\) it is the length of parallel channel, and the first term on the left represents the inertia pressure drop term, which is the additional pressure drop caused by the transient change of flow. The formula (5) can be sorted as:

$$ \begin{gathered} \frac{L}{{A_{i} }}\frac{{\partial m_{i} }}{\partial t} = \left( {P_{in} - P_{out} } \right) - \\ \Delta P_{a,i} - \Delta P_{f,i} - \Delta P_{el,i} \\ \end{gathered} $$
(6)

In the formula (6), \(\Delta P_{a,i}\) is the acceleration pressure drop in the \(i\) channel, \(\Delta P_{f,i}\) is the friction pressure drop in the \(i\) channel, and \(\Delta P_{el,i}\) is the gravity pressure drop in \(i\) the channel.

The equations (7) are the flow distribution calculation model of parallel channels.

$$ \left\{ \begin{gathered} \frac{L}{{A_{1} }}\frac{{\partial m_{1} }}{\partial t} = \left( {P_{in} - P_{out} } \right) - \Delta P_{a,1} - \Delta P_{f,1} \hfill \\ - \Delta P_{el,1} \hfill \\ \frac{L}{{A_{2} }}\frac{{\partial m_{2} }}{\partial t} = \left( {P_{in} - P_{out} } \right) - \Delta P_{a,2} - \Delta P_{f,2} \hfill \\ - \Delta P_{el,2} \hfill \\ \cdot \cdot \cdot \hfill \\ \frac{L}{{A_{i} }}\frac{{\partial m_{i} }}{\partial t} = \left( {P_{in} - P_{out} } \right) - \Delta P_{a,i} - \Delta P_{f,i} \hfill \\ - \Delta P_{el,i} \hfill \\ \frac{{\partial m_{t} }}{\partial t} = \sum\limits_{i = 1}^{N} {\frac{{\partial m_{i} }}{\partial t}} \hfill \\ \end{gathered} \right. $$
(7)

2.4 Model of Heat Transfer Between Assembles

The heat transfer model of thermal program assemblies developed in this paper adopts multi-layer heat conduction model, and the heat transfer coefficient between the m and n assembly of the j layer can be expressed as:

$$ \frac{1}{{K_{m,n}^{j} }} = \frac{\delta }{{k_{liq}^{j} }} + \frac{2t}{{k_{s}^{j} }} + \frac{1}{{h_{m}^{j} }} + \frac{1}{{h_{n}^{j} }} $$
(8)

In the formula (8): \(K_{m,n}^{j}\) is the total heat transfer coefficient between the assembly \(m\) and \(n\) in the \(j\) layer, \(\delta\) is the distance between the assembly boxes, \(k_{liq}^{j}\) is the thermal conductivity of the coolant between the boxes, \(t\) is the thickness of the assembly box, \(k_{s}^{j}\) is the thermal conductivity of the assembly box, \(h_{m}^{j}\) is the convection heat transfer coefficient between the coolant and the wall of the \(m\) assembly box in the \(j\) layer.

Considering the calculation efficiency, the core assembly is divided into several circles. Since the power distribution of assemblies in the same circle is relatively consistent, the heat exchange between assemblies in the same circle is ignored. This paper only considers the heat exchange between fuel assemblies, not between fuel assemblies and reflector assemblies.

3 Candle

The research object of this paper is based on the lead cooled traveling wave reactor [5] designed by Chinese Academy of Sciences. With reference to its core physical design, thermal hydraulic design and analysis are carried out on this basis. The radial and arrangement of the reactor core are shown in Fig. 1. Table 1 gives the basic parameters of traveling wave reactor. The fuel is metal fuel, the coolant is liquid metal lead, and the cladding is made of HT-9 material.

Fig. 1.
figure 1

Core Radial Arrangement

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Table 1. Reactor Core Parameters
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4 Result Analysis

4.1 Axial Heat Transfer

The core of lead cooled traveling wave reactor is divided into new fuel area, main combustion area and spent fuel area along the axial direction. The core heat is mainly generated by the main combustion area. The power density of new fuel area and spent fuel area is low, and the axial power density difference is great.

Based on the given power distribution, Fig. 2 compares the central temperature and coolant temperature of various types of fuels. The coolant inlet temperature is 650 K. Since the axial power distribution of the fuel assembly has been given, that is, the spatial heat source distribution of the coolant, the temperature change of the coolant in the axial direction is basically the same. Because the thermal conductivity of oxide fuel is less than that of nitride, and that of nitride fuel is less than that of metal fuel, the central temperature of oxide fuel is the highest, which can reach 2350 K, and that of metal fuel is the lowest under a given power distribution. Due to the low power density in the spent fuel area (0 < node number < 7), the temperature rise of the coolant in this area is slow. The power density in the main combustion zone (7 < node number < 14) is high, the coolant temperature rises rapidly, and the fuel has the highest central temperature in this zone. The power density of the new fuel area (14 < node number < 20) is almost zero, and the coolant temperature hardly rises.

Fig. 2.
figure 2

Temperature of Fuel and Coolant in the Core

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Figure 3 compares the calculated temperature difference of fuel rods before and after considering axial heat conduction for various types of fuels. The influence ratio is defined as:

$$ \Delta T/T = \frac{{\left( {T_{Axial\_heat\_conduction} - T_{Not\_axial\_heat\_conduction} } \right)}}{{T_{Axial\_heat\_conduction} }} $$
(9)

As can be shown in the Fig. 2, the middle section of the fuel rod will transfer heat to start and end of fuel. Therefore, the fuel temperature in the middle area (7 < node number < 14) has decreased, and the maximum decrease ratio is \({2.6\times 10}^{-4}\). It can be seen from the Fig. 3 that the metal fuel has a high thermal conductivity, so its axial heat conduction has a great effect, while the oxide fuel has a small effect on the axial heat conduction. The axial heat conduction of all types of fuels has a proportion of 10−5 − 10−4 on the temperature distribution.

Fig. 3.
figure 3

Temperature of Various Types of Fuel under the Same Condition

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Figure 4 compares the calculated temperature difference of coolant before and after considering axial heat conduction for various types of fuels. Axial heat transfer of the fuel rod also has a certain impact on the temperature of the coolant. As the fuel temperature in the spent fuel area (0 < node number < 7) rises slightly under the action of axial heat conduction, the heat transferred to the coolant there will increase. Therefore, the coolant temperature in this area rises faster under the consideration of axial heat conduction and is higher than the coolant temperature without consideration of axial heat conduction. However, the fuel temperature in the main combustion zone (7 < node number < 14) decreases slightly under the action of axial heat conduction, so the heat transferred to the coolant there will decrease. Therefore, under the consideration of axial heat conduction, the coolant temperature in this zone rises slowly and is lower than that without consideration of axial heat conduction. In the new fuel area (14 < node number < 20), the heat transferred to the coolant there will increase. Therefore, considering axial heat conduction, the coolant temperature in this area increases rapidly. Due to the high thermal conductivity of metal, the effect of axial heat conduction is slightly more obvious than that of other types of fuels. Therefore, the coolant using metal fuel has the largest temperature difference before and after considering axial heat conduction.

Fig. 4.
figure 4

Temperature of Coolant with Various Types of Fuel

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The difference of fuel temperature and coolant temperature between considering axial heat conduction and not is about 10−5 − 10−4, which has little impact on the thermal hydraulic calculation of the core. This study shows that its influence on the temperature distribution of fuel rod is very limited, and the influence of axial thermal conductivity can be ignored to a certain extent.

4.2 Flow Distribution

In order to further analyze the core flow distribution characteristics, a group of representative assemblies are selected along the core diagonal for research and analysis. Figure 1 has 14 assemblies from 186 to 162 along the diagonal, corresponding to the serial numbers from 1 to 14.

In this paper, the normalized flow is introduced, which is defined as the ratio of the inlet flow of each assembly in the core to the average inlet flow of the core assembly.

Fig. 5.
figure 5

Power distribution and flow distribution

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Figure 5 shows the radial distribution curve of core power and the flow distribution of each assembly before and after considering flow distribution. It can be seen from the Fig. 5 that after considering the flow distribution, the flow share of the middle assemblies of the core will increase to a certain extent. Among them, the power normalization coefficient of the hottest assembly is 1.37, and its flow normalization coefficient is 1.03. The flow share of core edge assemblies will decrease, and the normalized power coefficient of the coldest assembly is 0.66, and its flow normalized coefficient is 0.98. It can be seen that the core has a certain flow distribution capacity, but there are some limitations, that is, there are “overheating” of high-power assembly and “supercooling” of low-power assembly in the core.

Figure 6 shows the coolant outlet temperature distribution of assemblies along the diagonal of the core under rated working conditions. The coolant outlet temperature in the middle of the core is extremely high, reaching 917 K. The coolant outlet temperature gradually decreases outward along the radial direction of the core, and the coolant outlet temperature of the most marginal assembly is 780 K. After considering the flow distribution, the coolant outlet temperature of the assemblies inside the core have decreased to a certain extent. Among them, the coolant outlet temperature of the intermediate assemblies in the core has decreased by about 8 K, and the coolant temperature at the edge of the core has slightly increased. This is because the automatic distribution of the core flow makes the intermediate thermal assemblies distribute more coolant flow, while the edge assemblies distribute less coolant flow, as a result, the coolant temperature of the core middle assembly decreases and the coolant temperature of the edge assembly increases.

Fig. 6.
figure 6

Coolant temperature in the core outlet

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In order to analyze the core flow distribution characteristics, this paper studies the influence of the flow channel area of assembly on the core flow distribution capacity. In order to evaluate the mismatch between the core flow and power, the non-uniformity coefficient is introduced, and its formula is defined as:

$$ M = \frac{1}{N}\sum\limits_{1}^{N} {\left| {\frac{{f_{i,power} }}{{f_{i,flow} }} - 1} \right|} $$
(10)

\(N\) is the number of assemblies, \(f_{i,power}\) is the power factor of the \(i\) assembly, \(f_{i,flow}\) is the flow factor of the \(i\) assembly.

Fig. 7.
figure 7

Relation Curve Between Non-Uniformity Coefficient and ∆T

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Figure 7 shows the non-uniformity coefficient under different inlet and outlet temperature differences of the core. The inlet and outlet temperature of the core is changed by adjusting the total inlet flow of the core. When the inlet and outlet temperature difference increases, the power flow non-uniformity coefficient will decrease to a certain extent. When the inlet and outlet temperature difference of the core is small, as shown in the Fig. 7, the minimum temperature difference is 38 K, the non-uniformity coefficient is 0.209. When the temperature difference between the inlet and outlet of the core is large, as shown in the Fig. 8, the maximum temperature difference is 364 K, the non-uniformity coefficient is 0.201. This is because when the average inlet and outlet temperature difference of the core increases, the coolant temperature difference and density difference between the hot and cold assemblies will increase, that is, the driving force for automatic flow regulation is increased, and the flow distribution of the core is more suitable for the power distribution of the core.

4.3 Assembly Heat Transfer

The distribution of core flow is not considered in this subsection, and the basic assumption is that the coolant flow of each fuel assembly is consistent. Considering the symmetry of the core, a quarter of the core is taken as the research object. There are 7 layers of fuel assemblies in the radial direction of the reactor core. Selects 0, 12, 30, 54, 84, 120 and 162 assemblies in Fig. 1 as the research objects, which respectively correspond to No. 1–7 in the Fig. 8. Figure 8 compares the coolant outlet temperatures of the above assemblies before and after the assemblies’ heat exchange is considered. After considering the heat exchange, the coolant temperature in the hot channel decreases from 918.69 K to 916.01 K, decreases by 2.68 K, increases from 778.31 K to 787.25 K, increases by 8.94 K, and the maximum coolant temperature difference at the channel outlet decreases by 11.62 K. However, the influence of assembly heat transfer on the coolant temperature distribution is limited.

Figure 9 shows the three-dimensional distribution of the heat flux transferred between the core assemblies. Considering the symmetry of the core, a quarter of the core is taken as the research object. It can be seen from the Fig. 9 that in the radial direction, the heat exchange power of the assemblies in the middle area of the core is small, and the heat exchange power gradually increases along the radial direction. This is because the difference between the normalized power of adjacent assemblies inside the core is small, so the coolant temperature difference between adjacent assemblies in this area is small, so the heat exchange power between assemblies inside the core is small. The power density of heat transfer between core edge assemblies is large, and the heat flux density at the radial edge is about 4–5 times that in the middle region.

Fig. 8.
figure 8

Coolant Outlet Temperature Between Heat Transfer and Not Heat Transfer

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Fig. 9.
figure 9

C Three-Dimensional Distribution of Heat Transfer Power

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5 Conclusion

Thermal-hydraulic program is developed in this paper. The thermal hydraulic program developed is used to calculate the core thermal characteristics of lead cooled traveling wave reactor. The research results show that:

  1. 1)

    The influence of axial heat conduction on fuel temperature and coolant temperature distribution is compared and analyzed by using the calculation program. The calculated axial thermal conductivity is about 10−5 − 10−4, so its influence on the temperature distribution of fuel rod is very limited, and the influence of axial thermal conductivity can be ignored.

  2. 2)

    The developed thermal hydraulic program is used to study the flow distribution characteristics of the whole reactor core. The calculation results show that the reactor core has capacity of automatic flow distribution. At the same time, this paper studies the influence of inlet and outlet temperature difference on the core flow distribution capacity. When the inlet and outlet temperature difference of the core is large, the core has a large flow automatic distribution capacity. However, there is still a mismatch between the normalized flow and the normalized power of the core, that is, it is necessary to optimize the core flow distribution in the future.

  3. 3)

    The program was used to carry out the heat exchange calculation of the whole core assemblies. After considering the assembly heat exchange, the temperature difference at the coolant outlet between each channel decreased, from 140.38 K to 128.76 K, the change was relatively small. This is because the heat resistance of heat exchange between assemblies is large, and the coolant temperature difference in adjacent assemblies is not very significant. Therefore, the assembly heat exchange capacity is weak under this working condition.