Keywords

1 Introduction

Compared with the traditional rod-type fuel, the plate-type fuel has obvious advantages, which is mainly achieved in the higher heat flux under the same volume, which also means that the plate-type fuel reactor has higher power under the same volume. From the perspective of thermal-hydraulics, the flow channel of plate-type fuel is a narrow rectangular channel, its size is usually between 1 mm and 3 mm, which has a certain strengthening effect on convective heat transfer, and its heat transfer efficiency is higher than that of rod-type channel. In addition, plate-type fuel can obtain higher burn up depth than rod fuel. In conclusion, the application of plate fuel improves the economy of the reactor.

The earliest use of plate-type fuel is MTR reactor [1] (material test reactor), which uses U-Al alloy plate-type fuel. After that, MIT built MITR reactor for the purpose of research and education [2], and Indonesia built the reactor RSG-GAS for material testing and radioisotope production [3]. China advanced research reactor (CARR) is a pool reactor, which is cooled by light water and moderated by heavy water. These reactors have the same characteristics, that is, to increase the heat transfer area between the fuel plate and the coolant as much as possible by arranging ribs on the fuel plate and using curved fuel plate.

From the above application of plate-type fuel, it can be seen that the use of plate-type fuel as reactor fuel is mainly due to the compact core structure and high power density of plate-type fuel. Therefore, the research on thermal-hydraulics of plate-type fuel mainly focuses on how to take more heat away. Generally, the coolant channel in plate-type fuel is rectangular channel, which belongs to narrow channel. Therefore, a lot of research work has been done on the flow and heat transfer characteristics of narrow rectangular channel.

1.1 Research Status of Flow and Heat Transfer in Rectangular Channel

The flow and heat transfer characteristics of narrow rectangular channel is one of the concerns of plate-type fuel research. The research on it can provide reference and verification for the design of plate-type fuel element. Generally, for plate-type fuel elements, the channel shape is narrow rectangular channel, and the channel gap is generally 2–3 mm, some even less than 2 mm.

In terms of flow resistance, Ha Taesung of McMaster University studied the velocity distribution and friction resistance characteristics in the flow passage when coolant enters the fuel assembly from the upper part [4]. The experimental results show that the inlet flow distribution is uniform in different channels, and the maximum difference is less than 5%; In a single channel, the velocity in the middle of the channel is the largest, and then slowly attenuates, and rapidly attenuates to 0 near the outside of the channel. Through the calculation of pressure drop in the channel, the results show that the conventional friction resistance relationship is also suitable for rectangular channel.

In terms of flow pattern, Chang of Shandong University measured the gas-liquid two-phase flow pattern on the heating experimental bench with rectangular channel width of 0.5 mm to 2.0 mm respectively. The results show that the void fraction is about 0.7 in the transition from isolated or confined bubble to plug flow and about 0.9 in the transition from plug flow to annular flow. Wang observed the flow pattern characteristics of hydrodynamic boiling in vertical narrow rectangular channel under the conditions of low pressure and inlet temperature supercooling. Four flow patterns were observed: diffuse bubble flow, combined bubble flow, agitated flow and annular flow. The experimental data were compared with the existing typical flow pattern diagrams. The results show that the flow pattern of heating steam-water and its transformation law are obviously different from that of adiabatic air-water.

In terms of heat transfer, Sudo used JRR-3 reactor fuel elements to carry out experimental research on single-phase heat transfer in rectangular channel [5]. By comparing the experimental results, it can be seen that the Dittus-Bolter, Sider-Tate and Colburn relations are suitable for single-phase heat transfer in rectangular channel, but the relations used in laminar flow are different from those used in circular tube, and the Nusselt numbers of upflow and downflow are also different. Qiu of Xi'an Jiaotong University studied the characteristics of saturated boiling heat transfer in horizontal rectangular channel. The experimental results show that the heat transfer coefficient of rectangular channel is significantly higher than that of circular tube in laminar and turbulent regions. When the gap size is less than 1.5 mm, the heat transfer effect is obviously weakened, while when the gap size is more than 1.5 mm, the gap size has no obvious effect on the heat transfer effect. Tian studied the heat transfer characteristics of rectangular channel under natural convection. The results show that in the case of natural circulation convective heat transfer, the decrease of inlet subcooling has a weakening effect on heat transfer, and the heat transfer capacity will be improved with the increase of heating power. Moreover, the experimental results show that the Gnielinski relation [6] can be used to predict the heat transfer coefficient under natural circulation.

Through the investigation of the flow and heat transfer characteristics of rectangular channel, it can be concluded that the flow and heat transfer in rectangular channel is different from the traditional round tube and rod bundle, mainly in the enhancement of heat transfer in narrow channel.

1.2 Research Status of Plate-Type Fuel Code Development

In the reactor design stage, the design parameters often need to be calculated with the help of thermal-hydraulic analysis code, and the simulation code can also verify the existing design. The thermal hydraulic code can also be used to study specific phenomena and provide direction for experimental research, Therefore, the development of thermal analysis code for plate-type fuel reactor plays an important role in reactor construction and academic research. In the world, many countries, including China, have independently developed codes suitable for plate-type fuel. The developed codes are shown in Table 1.

Table 1. The existing thermal hydraulic analysis code for plate-type fuel
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In the design stage, the thermal-hydraulic analysis code is used to obtain the design parameters and predict the state of the reactor, so as to verify, evaluate and modify the existing design. The commonly used thermal-hydraulic analysis methods include computational fluid dynamics method, single-channel analysis method and subchannel analysis method. Although computational fluid dynamics (CFD) can obtain fine parameter distribution, it is difficult to simulate the reactor scale, and the two-phase calculation needs to be strengthened; Although the single-channel analysis method has a small amount of calculation, it is unable to obtain the distribution of core thermal parameters; The subchannel analysis method considers that there are transverse momentum, mass and heat mixing in the process of coolant flow between adjacent channels. Therefore, compared with the single-channel model, the calculation results are closer to the real parameters, and the thermal parameters of the core can be obtained with less calculation.

According to the literature research, at present, most of the international plate-type fuel analysis codes use single-channel models to verify and analyze accident conditions. For the subchannel model, there is only the MATRA-h subchannel analysis code developed by South Korea on the basis of MATRA, and there is no subchannel analysis code for plate-type fuel in China. However, under the requirements of refined analysis and under the background of nuclear power software autonomy, the development of subchannel analysis code suitable for plate-type fuel can obtain more accurate and refined thermal hydraulics than single-channel models. The results of the analysis are also helpful for the verification and analysis of the experimental research reactors that have been built and are in the design stage in China. For the above purposes, it is necessary to develop the best estimation code for the analysis of plate-type fuel subchannels with independent property rights.

2 Development of Plate-Type Fuel Subchannel Code

The code reads in the geometric structure information, boundary conditions, control information and other parameters needed for the calculation in the form of an input card. In the initialization phase, the read-in data is used to construct a geometric model and set boundary conditions. After the initialization is completed, the solution process of the code starts. First, use the initial input mass flow to perform transient calculations, and then redistribute the inlet flow after the transient calculations are completed, until the pressure drop of each subchannel is in a balanced state. In each transient calculation, the calculation is ended by judging whether the steady-state convergence condition is reached or the calculation time is reached. In transient calculations, discrete conservation equations are solved by the SIMPLE algorithm. After the transient calculation is completed and the pressure drop of the subchannel is balanced, the calculation result will be output. The calculation process of the code is shown in Fig. 1.

2.1 Basic Conservative Equations

The conservation equations coded are presented in this section. The basic equations of the mathematical model are derived by applying the general equations of continuity, energy, and momentum. The equations are:

Mass:

$$\frac{\partial }{\partial t}\left( {\alpha_{k} \rho_{k} } \right) + \nabla \cdot \left( {\alpha_{k} \rho_{k} \overrightarrow {V}_{k} } \right) = L_{k} + M_{k}^{T}$$
(1)

The first term at left is the change in mass of each phase in the control body over time, and the second term at left is the convection term.

Momentum:

$$\begin{gathered} \frac{\partial }{\partial t}\left( {\alpha_{k} \rho_{k} \overrightarrow {V}_{k} } \right) + \frac{\partial }{\partial x}\left( {\alpha_{k} \rho_{k} u_{k} \overrightarrow {V}_{k} } \right) + \frac{\partial }{\partial y}\left( {\alpha_{k} \rho_{k} v_{k} \overrightarrow {V}_{k} } \right) + \hfill \\ \frac{\partial }{\partial z}\left( {\alpha_{k} \rho_{k} w_{k} \overrightarrow {V}_{k} } \right) = \alpha_{k} \rho_{k} g + \nabla \cdot \left[ {\alpha_{k} \left( {\tau_{k}^{{}} + T_{k}^{{}} } \right)} \right] - \hfill \\ \alpha_{k} \nabla P + \overrightarrow {M}_{k}^{L} + \overrightarrow {M}_{k}^{d} + \overrightarrow {M}_{k}^{T} \hfill \\ \end{gathered}$$
(2)

The first term at left is the change of momentum in the control body over time, the second, third, and fourth terms are momentum convection; the first term at right is gravity, the second is pressure, and the third is viscous force and turbulent shear force.

Energy:

$$\begin{gathered} \frac{\partial }{\partial t}\left( {\alpha_{k} \rho_{k} h_{k} } \right) + \nabla \cdot \left( {\alpha_{k} \rho_{k} h_{k} \overrightarrow {V}_{k} } \right) = \hfill \\ - \nabla \cdot \left[ {\alpha_{k} \left( {\overrightarrow {Q}_{k} + \overrightarrow {q}_{k}^{T} } \right)} \right] + {\kern 1pt} {\kern 1pt} {\kern 1pt} \Gamma_{k} h_{k}^{{}} + q_{wk}^{{}} + \alpha_{k} \frac{\partial P}{{\partial t}} \hfill \\ \end{gathered}$$
(3)

The first term at left is the change of energy in the control body over time; the second term is the energy convection term. The first term at right side of the equation is the heat exchange caused by heat conduction between phases, turbulence mixing and cavitation drift, the second term is the energy change caused by the phase change; the third term is the energy entering the fluid due to wall heating; The fourth item is pressure doing work.

Fig. 1.
figure 1

Evaluation scheme of the subchannel code

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2.2 Constitutive Models

The wall heat transfer model determines the heat transfer coefficient between fluid and fuel, and directly affects the temperature distribution in the whole core. According to the relationship of void fraction, enthalpy and temperature, the heat transfer modes can be divided into single-phase liquid heat transfer, subcooled boiling, saturated boiling, transitional boiling, annular boiling, dispersed flow and single-phase steam heat transfer.

For single-phase fluid heat transfer, the relationship proposed by Dittus-Boelter [2, 7] is used:

$$Nu = 0.023{\text{Re}}^{0.8} \Pr{^{0.4}}$$
(4)

For subcooling and saturated nucleate boiling, use the relationship proposed by Chen [8]:

$$h_{Chen} = h_{fc} + h_{nb}$$
(5)
$$h_{fc} = 0.023{\text{Re}}^{0.8} \Pr{^{0.4}} \left( {\frac{{k_{l} }}{{D_{e} }}} \right)F$$
(6)
$$h_{nb} = 0.00122\left( {\frac{{k_{l}^{0.79} C_{pl}^{0.45} \rho_{f}^{0.49} }}{{\sigma_{l}^{0.5} \mu_{f}^{0.29} H_{fg}^{0.24} \rho_{g}^{0.24} }}} \right)\Delta T_{sat}^{0.24} \Delta P_{sat}^{0.75} S$$
(7)

For the heat transfer coefficient after transitional boiling, the heat transfer effect of four parts should be considered: convective heat transfer on the wall qwv, boiling heat transfer on the liquid qwb, radiative heat transfer on the wall qrwl and radiative heat transfer on the wall qrwv.

$$q = q_{wv} + q_{wb} + q_{rwv} + q_{rwl}$$
(8)

In particular, under the annular flow, the modified Bromley relationship is used to calculate the heat flux [9]

$$q = 0.62\left( {\frac{{D_{e} }}{\lambda }} \right)^{0.172} \left[ {\frac{{k_{g}^{3} \rho_{g} (\rho_{f} - \rho_{g} )H_{fg} G}}{{D_{e} \mu_{g} (T_{w} - T_{sat} )}}} \right]^{0.25} (T_{w} - T_{sat} )$$
(9)

In forced convection heat transfer, when the heat flux density is so high that it is difficult for the liquid phase to wet the heating surface, the peak heat flux density at this time is called the Critical Heat Flux (CHF). This paper uses the Sudo relationship [10] to determine the occurrence of CHF.

$$q_{CHF1}^{*} = 0.005\left| {G^{*} } \right|^{0.611}$$
(10)
$$q_{CHF2}^{*} = \left( {\frac{A}{{A_{H} }}} \right)\Delta T_{sub,in}^{*} \left| {G^{*} } \right|$$
(11)
$$q_{CHF3}^{*} = 0.7\left( {\frac{A}{{A_{H} }}} \right)\frac{{(w/\lambda )^{0.5} }}{{\left( {1 + \left( {\frac{{\rho_{g} }}{{\rho_{f} }}} \right)^{0.25} } \right)^{2} }}$$
(12)
$$q_{CHF4}^{*} = 0.005\left| {G^{*} } \right|^{0.611} \left( {1 + \frac{5000}{{\left| {G^{*} } \right|}}\Delta T_{sub,ex}^{*} } \right)$$
(13)

For ascending flow, if qCHF1 <= qCHF3, then qCHF = qCHF3. If qCHF1 <= qCHF2, then qCHF = qCHF4. In other cases, qCHF = qCHF1.

For descending flow, if qCHF2 <= qCHF3, then qCHF = qCHF3. If qCHF1 <= qCHF2, then qCHF = qCHF4. In other cases, qCHF = qCHF2.

For the flow pattern before CHF, since it is assumed that the vapor phase does not contact the wall surface, the friction force between the vapor phase and the wall surface is set to zero. After CHF, the Darcy friction resistance coefficient can be calculated using the following formula:

$$f_{w,k} = \max \left\{ {\begin{array}{*{20}c} {{(96}.0/{\text{Re}} {\kern 1pt} {\kern 1pt} ){\kern 1pt} (\mu_{w} /\mu_{k} )^{0.14} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {0.316{\text{Re}}^{ - 0.25} } \\ \end{array} } \right.$$
(14)

2.3 Flow Distribution Model

The traditional flow distribution method using fixed step iterations judges whether the balance is reached by the difference between the maximum pressure drop and the average pressure drop in the subchannels. If the balance is not reached, the flow needs to be redistributed. This method requires multiple iterations, which greatly increases the calculation time.

Therefore, a new flow distribution method is proposed to distribute the flow of each subchannel. This method regards the subchannel pressure drop as a linear function of the flow rate, and predicts the subchannel inlet mass flow rate for the next flow iteration by means of a fitting relationship. The function is as follows.

$$\Delta P_{i}^{n} = a^{n} + b^{n} W_{i}^{n}$$
(15)

In the calculation, the linear function of the iteration step is solved for each subchannel, and a new mass flow rate is obtained from the average pressure drop, as follows:

$$W_{i}^{n + 1} = \frac{{P_{ave}^{n} - a^{n} }}{{b^{n} }}$$
(16)

2.4 Special Heat Conduction Treatment of Side Cladding

Since the plate-type fuel has a cladding on the side, as shown in Fig. 2, if the closed subchannel is calculated, the heat conduction of this part of the cladding does not need to be considered. If the opened subchannel is calculated, the heat conduction of the side shell for the subchannel needs to be calculated, otherwise the energy will not be conserved.

The gray part in Fig. 2 is the side cladding. For the side cladding, after solving the heat conduction of the pellets (using the additional source term method), the heat entering the side cladding is obtained explicitly.

Fig. 2.
figure 2

Diagram of side cladding

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$$T_{sc} = \frac{{\sum\limits_{i = 1}^{n} {b_{i} R_{i} } + q_{chn} }}{{\sum\limits_{i = 1}^{n} {R_{i} } + h_{chn} }}$$
(17)

For the nodes in the core block that are in contact with the side bread shell, use the additional source term method to subtract the heat derived from the side bread shell. The source term is as follows:

$$b = q\Delta x + a(T_{sc} - T_{i} )$$
(18)

3 Verification of Plate-Type Fuel Subchannel Code

The State Key Laboratory of Multiphase Flow at Xi’an Jiaotong University carried out experiments on the flow resistance and heat transfer performance of the double rectangular narrow slits between the plate-type fuels. The overall length of the narrow slit part of the test piece is 1200 mm, the effective heating section length is 900 mm, and the effective pressure measurement length is 1000 mm; the entrance has a stable section of 150 mm, the exit has a stable section of 50 mm, and the plate thickness is 0.7 mm. The cross-sectional size of the circulation channel is 50 mm × 1.8 mm; the processed test piece is measured by a spiral micrometer, and its average gap width is 1.806 mm, which does not exceed 0.4% of the design value. Use the developed plate-type fuel subchannel code to simulate the above experiment.

In order to verify the influence of the mass flow rate on the heat transfer coefficient, the method of variable-controlling is adopted. The experimental conditions are shown in Table 2. The mass flow rates are 600, 800, and 1000 kg·m−2·s−1.

Table 2. Boundary condition
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The developed plate-type fuel subchannel code is used to calculate the heat transfer coefficient, which is divided into 90 nodes along the axial direction. The comparison between the heat transfer coefficient along the axial position and the experimental results is shown in Fig. 3.

Fig. 3.
figure 3

Comparison between calculated and experimental values of heat transfer coefficient under the influence of mass flow rate

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The heat transfer area calculated in Fig. 3 includes the stages from single-phase heat transfer, subcooled boiling to saturated boiling. There is a sudden jump in the calculated value of the subchannel code, which is caused by the switch of the heat transfer relationship from subcooled boiling to saturated boiling. In the single-phase liquid heat transfer phase, the heat transfer coefficient calculated by the code is in good agreement with the experimental data; in the subcooled boiling and saturated boiling phases, although the heat transfer coefficient calculated by the code has a certain difference with the experimental data, the trend is still in good agreement. It can be seen from Fig. 3 that as the mass flow rate increases, the position where the saturation point appears gradually moves upward.

4 Subchannel Analysis of a MTR Bundle

In this section, the MTR reactor is used as a steady-state analysis and verification model. The schematic diagram of the fuel assembly is shown in Fig. 4.

There are 25 fuel assemblies in the MTR core, each of which contains 23 fuel plates placed at equal intervals; there are 4 control body assemblies, each of which contains 17 fuel plates.

Figure 4 shows part of the structure of the standard assembly. The geometric parameters of the flow channel and fuel plate are shown in Table 3. The physical parameters used in the calculation are shown in Table 4.

Fig. 4.
figure 4

The structure of the plate-type fuel in MTR reactor

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Table 3. Geometry of the fuel
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In the calculation, it is assumed that the body heat sources in the fuel plate are uniformly distributed, and the power along the axial direction is considered to be a cosine distribution.

Song from Harbin Engineering University calculated the outlet temperature of MTR reactor assembly by FLUENT. The calculation area used is shown in red area in Fig. 4. In the red area, there are two fuel plates and three subchannels from top to bottom. The calculation results of the developed subchannel code are compared with the calculation results of FLUENT, and the results are shown in Table 5.

Table 5 shows that under the same conditions, the relative error between the results of the developed subchannel code and that of FLUENT is less than 0.05%, which is in good agreement with each other. The difference between the results of the subchannel code and that of the CFD is mainly due to the fact that the heat conduction of the fuel plate is not considered in the CFD, so there is a little difference in the heat distribution, but the total energy is conserved.

Table 4. Parameters related to MTR
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Table 5. Outlet temperature of the channel calculated by CFD and subchannel code
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Amgad Salama [11] of South Korea used the subchannel code MATRA to calculate the axial variation of coolant and cladding temperature in MTR reactor under steady-state conditions. The calculation area used by Salama is the yellow area in Fig. 4, which is a quarter of the flow channel.

The calculation model used in the developed code is shown in Fig. 5. It contains a complete fuel plate and two flow channels. The width of the flow channel is half of the actual width, which is equivalent to four times of the calculation area of Amgad Salama. During the modeling process, the single channel is divided into three subchannels, as shown in Fig. 5.

The results of cladding temperature distribution and coolant temperature distribution are shown in Fig. 6. The results of coolant temperature and fuel cladding temperature are the same due to the same heating conditions and flow rate of the three subchannels divided by the same channel, so the calculation results of subchannel 2 were used.

Fig. 5.
figure 5

Diagram of subchannel code modeling

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Fig. 6.
figure 6

Temperature distribution along the axis

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Figure 6 shows that the temperature distribution of the cladding is very close between MATRA and the developed subchannel code under the approximate operating conditions. In Fig. 6, the coolant temperature predicted by Matra is slightly lower than the developed subchannel code, which may be due to the difference in the calculation of total power estimation. In this example, because the pressure difference between the subchannels is the same, there is almost no transverse flow between the subchannels. In conclusion, the developed subchannel code can be used for thermal hydraulic analysis of plate-type fuel reactor.

5 Results

Plate-type fuel is widely used in different reactors due to its high heat transfer efficiency. Its fuel shape and flow path structure are different from those of traditional pressurized water reactor, which leads to different thermal hydraulic phenomena and temperature field distribution in fuel. In order to obtain the thermal-hydraulic characteristics in the rectangular channel, this paper conducts related research on the thermal-hydraulic characteristics in the rectangular channel, and develops a subchannel analysis model and code suitable for plate-type fuel reactors.

This article first fully investigated the flow heat transfer, friction model, plate-type fuel heat conduction model and flow distribution model suitable for rectangular flow channels. Based on the existing two-fluid three-flow field subchannel code, a subchannel analysis code suitable for plate-type fuel reactors has been developed. After that, the comparison with the flow and heat transfer experiment in the rectangular channel shows that the code can accurately predict the heat transfer coefficient under the rectangular channel. The comparison and verification with the simulation calculation results of the MTR reactor show that the developed subchannel code can simulate the thermal-hydraulic phenomena of the plate-type fuel assembly.

According to previous studies, the parametric uncertainty such as theoretical and experimental analysis uncertainties, instrumentation and control inaccuracies, manufacturing tolerances, material properties and correlation uncertainties have effect on the hot channel, so in the further study, the parametric uncertainty will be analyzed to make the analysis more realistic.