A new method for characterizing the heat transfer capacity of the divertor and high heat flux components

The divertor is usually exposed to extreme incident heat flux conditions in nuclear fusion engineering and its ability to remove the heat is extremely crucial for the safe operation of magnetic confinement nuclear fusion device. The evaluation of the heat removal capacity of the divertor has been always based on high heat flux (HHF) tests. Yet such methods are only available for the small size of experimental components, it is more difficult to compare under different loading and cooling conditions. In order to assess the heat removal capacity in real time and accurately, a new method was developed to characterize the heat removal capacity of the divertor and other high heat load components based on the cooling water thermal decay time constant, the time constants model has also been established. The thermal decay time constant was obtained by fitting the characteristic curves of the water temperature differences with time during the cooling phase of different components under HHF tests. This index is used to quantitatively analyze, characterize and evaluate the heat transfer capacity of the system of the divertor.


Introduction
The divertor is one of the most significant plasma-facing components (PFCs) within the vacuum vessel of the tokamak device, which is subjected to very harsh operating conditions with high steady and transient high heat flux (HHF) loading. Therefore, in order to ensure safe and reliable operation of the divertor, it is extremely necessary to evaluate its heat * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. removal capacity in real time and accurately. Nevertheless, the evaluation of the heat removal capacity of divertors has been mainly relying on the HHF tests for a long time. HHF tests are generally only suitable for the small size experimental components, which make it difficult to test the overall structure of the divertor. It is also hard to evaluate comparatively under different loading and cooling conditions, and even more impossible to judge the degradation of the performance during their service period. In addition, finite element analysis is also a convenient method for evaluating the heat transfer performance of HHF components. But it is mainly aimed at designing models and it is difficult to evaluate actual components. Therefore, it is extremely important to establish a rapid, reliable and real-time method for evaluating the heat transfer capability of both the divertor and HHF components.
Due to the thermal decay time constant is an inherent property, which is only related to material and construction rather than to loading and cooling conditions, so it can be used to characterize or evaluate the heat removal capacity of divertor and high heat load components. In general, the system has a smaller value of thermal decay time constant indicating better heat transfer performance.
The time constant approach has been applied in the field of fusion engineering. Mitteau et al proposed thermal time constant to characterize the delay before the thermal steady state, which allows the state of the component to be assessed. It is based on the observation of the surface temperature of the toroidal pumping limiter (TPL) in Tore Supra [1][2][3][4]. Higher thermal time constants in degraded structures compared to intact structures. However, the monitoring through the surface temperature is influenced by many parameters like additional thermal resistance in the structure or surface dust and layers. Their evolution in time complicates the thermal analysis. Mitteau et al simulated the thermal behaviors of tiles with junction defect tiles and a deposition layer and introduces a modified time constant method for analysis. He found that the deposited tiles showed a fast-decay stage during the decay phase [5]. According to the analyzed results, the deposited tiles can be distinguished from junction defect tiles by the slow-decay time constant, which is independent on the thermal resistance and the thickness of the deposition layer. He also studied the evolution of defects in two TPL tiles by analyzing infrared images obtained from experiments from 2006 to 2010 (about 10 000 plasma discharges), using changes in surface temperature with thermal time constants [6]. Whereas these applications characterize the condition of a component based on the rate of change of the time constant of its surface temperature, this paper characterizes the heat removal capacity of a bias filter and its high thermal load components by the value of the time constant of thermal decay of the cooling water.
In the current work, through fitting response curves to cooling water temperature response data during the cooling phase of the component under HHF tests, we obtained the thermal decay time constants. Different structures have different thermal decay time constants. The results indicate that it is successfully used to evaluate the heat transfer performance of various types of structural high heat load components. It can be used not only to quickly and accurately assess the heat transfer performance of divertors, but also to assess the extent of defects in components with HHF components. The method is highly applicable and has a wide range of applications. This paper is organized as follows. Section 2 introduces the theory and calculation of time constants, including theoretical derivation, experimental platform, acquisition of data, and the methods of analysis. In section 3, the research subjects are described in detail and calculations of experimental curve fitting for different structures are presented, demonstrating the feasibility of the method. As an example, the no-defects mono-block divertor mockup was investigated for the effect of heat flow density size, heat flux loading and cooling time on its thermal decay time constant. Section 4 discusses and analyzes the results obtained for the different research subjects. Finally, conclusions are presented in section 5.

Time constant fundamental principles
According to Fourier's law of heat transfer, the onedimensional differential equation for the thermal conductivity of a heat-conducting rod, using figure 1 as an example, is Here The initial condition is where the function f is given, and the boundary condition Try to find a solution of (1) that does not satisfy boundary condition (3), but has the following property: T is a product where the dependence of T on x, t is separated, that is: Simplifying the one-dimensional heat transfer equation using the separation of variables method, the following equation is obtained: Since the right-hand side depends only on x and the lefthand side only on t, both sides are equal to some constant value −λ. Thus: and It must be the case that λ < 0, there exist real numbers A, B, C such that and From (3) get C = 0 and that for some positive integer n, Therefore, the thermal decay time constant is From the equation (11): the thermal decay time constant τ is characteristic of the research object, i.e., it depends on factors such as structural forms and the properties of the materials (heat capacity, size, geometry) rather than on the magnitude and time of the loaded heat flux.

Time constant fitting method
In practice, the situation is complicated and instead of applying equation (11) directly, the time constant τ is obtained by fitting the temperature increase data of the cooling water during the cooling process of the HHF tests. HHF experimental platform basically includes an electron beam scanning heating system and its power control system, the vacuum room and vacuum maintenance system, the movable platform, cooling water system, temperature measurement (thermocouple, pyrometer) and data collecting system, etc. As shown in figure 2, the experimental model was installed on the mobile platform and the electron beam rapidly scans its surface to ensure uniform loading of the heat flux, creating a steady state thermal load. The surface temperature of the sample was measured by the Pyrometer (range of 350 • C-2000 • C). Two fastresponding and high-precision K-type thermocouples (range of 0 • C-800 • C) were used to measure the cooling water inlet temperature T 1 and outlet temperature T 2 in real time. The flow rate at the outlet was collected via flowmeter. The absorbed heat flux was calculated from the formula: where Q abs is the absorbed heat flux (MW m −2 ); c is the specific heat of water (J kg −1 K −1 ); G is the mass flow of cooling water (kg h −1 ); S is the loaded area (mm 2 ); ∆t is the equilibrium temperature rise of the water under the heat load ( • C); ρ is the water density (kg m −3 ); q is the flow rate (m 3 h −1 ); T 1 is the inlet temperature ( • C); T 2 is the outlet temperature ( • C). The loading conditions for the HHF tests were normally set as 15 s heating and 15 s cooling alternately to ensure that thermal equilibrium is adequately established during the heating and cooling process, which also referred to technical requirements of the ITER qualification program (typically 10 s ON and 10 s OFF).
During the HHF tests, the electron beam scans the test model rapidly and the temperature of the surface rises fast to reach a steady state, which was defined as the heating process. The electron beam stops scanning and the deposited heat reaches the cooling channel by heat conduction and is removed by the coolant causing the surface temperature to drop, defined as the cooling process.
It has been discovered that the temperature of the cooling water varies cyclically as the heat flux was alternately loaded in the experimental process. During the heating phase, the outlet water temperature (T 2 ) is gradually increasing and then reaches an equilibrium state, then after stopping the loading of the heat flux, the outlet water temperature (T 2 ) starts to decrease and reaches an equilibrium state. In the cooling phase, the water temperature change (∆T = T 2− T 1 ) follows an exponential decay pattern. The water cooling curve is fitted with a single exponential function to obtain the thermal decay time constant, as shown in equation (13): where is a function of the temperature difference between the outlet and the inlet with time ( • C), T (t) = T 2 − T 1 ; T max is the initial temperature difference of the cooling water at the start of the drop ( • C); t is cooling time (s); T 0 is the temperature at which the cooling water cools down to equilibrium ( • C); τ is the thermal decay time constant (s). The time constant of the fitted single exponential curve is the time constant of the experimental component.

Research subjects
In this paper, in order to investigate the feasibility and accuracy of the thermal decay time constant method in characterization, the main structure of PFCs structures for the mono-block and flat-type were investigated. The study is not only aimed at the research subjects are not only flat-type with different plasma facing materials (PFM) thicknesses L and different PFMs but also W/Cu mono-block plasma-facing units (PFUs) with various degrees of defects. The flat-type divertor mockup has a sandwich structure as shown in figure 3(a), which is composed of a rectangular external section W or KW (PFM) with CuCrZr (heat sink material) and 316 l stainless steel (structural material) forming a composite heat-sink connection, with oxygenfree copper as the intermediate transition layer. Details of the different flat-type models are listed in table 1. All PFCs of the flat-type manufactured have passed non-destructive testing (NDT) and interface have good bonding quality. The structure of W/Cu PFUs is shown in figure 3(b), where the tungsten is manufactured to the CuCrZr heat sink cooling water tube by technological combination of hot isostatic pressing (HIP) and hot radial pressing techniques, with a transition layer of Cu between them. Bonding interface quality was classified as no defects, minor and major defects depending on the   results of NDT. The PFUs that meet the acceptance criteria are defined no-defects PFUs. The PFUs with a small number of 2 × 3 mm defects are defined as minor defects PFUs (total area less than 10% of the inspection area), and the PFUs with large number of 2 × 3 mm defects are major defects PFUs [7]. It is worth noting that the main objective of this study was to investigate the feasibility of thermal decay time constants in evaluating the heat transfer performance of defective PFCs. Therefore, only three representative defect cases were chosen, and the effects of defect location and defect shape were not considered. The experimental data for the outlet temperature T 2 and inlet temperature T 1 were measured by thermocouples in the cooling system of the HHF experimental platform. The difference in water temperature of were obtained by subtracting the inlet temperature (T 1 ) from the outlet temperature (T 2 ) (T(t) = T 2 −T 1 ). The water temperature difference variation curve of no-defects PFUs under different HHF magnitudes were fitted separately by using equation (13). Figure 4 illustrates evolution of the cooling water temperature variation with time for no-defects PFUs under different HHF magnitudes and its fitted curve during the cooling phase. It is reasonable to observe that, with increasing HHF magnitude, T max rises from 0.92 • C (6 MW m −2 ) to 1.31 • C (10 MW m −2 ) under the same cooling conditions, which corresponds to equation (12). The thermal decay time constant of no-defects PFUs at heat flux of 6-10 MW m −2 were obtained from the fitting calculations and the results are shown in table 2.

Experimental data fitting of the mono-block and flat
The R 2 value can be used to assess how well the fit is working. When the R 2 value of the fitted curve is closer to 1, the (i.e., inlet velocity: 7 m s −1 ), inlet temperature: 20 • C. The outlet temperature T 2 and inlet temperature T 1 for three modes with no defects were measured by thermocouples in the cooling system of the HHF experimental platform. The difference in water temperature for each mode is obtained by subtracting the inlet temperature (T 1 ) from the outlet temperature (T 2 ) (T(t) = T 2 -T 1 ). The water temperature difference variation curve of no-defects PFUs under different loading and cooling modes were fitted separately by using equation (13). Figure 5 illustrates evolution of the cooling water temperature difference with time for no-defects PFUs under the three modes and its fitted curve during the cooling phase. The  Figure 6 illustrates the temperature evolution curves of the HHF loading center of the no-defects PFUs under different HHF magnitudes during screening tests and at different loading modes with the heat flux of 8 MW m −2 , which were measured using the pyrometer. The results indicate that the surface temperatures of the HHF loading area can quickly reach a steady state once the incident heat flux is exerted and can be quickly cooled down to a steady state by the cooling loop once the incident heat flux is stopped in loading cycle. It is    reasonable to observe that, with increasing HHF magnitude, the surface temperature rises from 698 • C (6 MW m −2 ) to 993 • C (10 MW m −2 ) under the same cooling conditions. The surface temperature of the no-defects PFUs can be kept steady in three loading modes. The minor-defects PFUs was operated at the heat flux of 7.3 MW m −2 in Mode II and Mode III. The cooling water conditions are set in the same conditions as for no-defects PFUs. The outlet temperature and the inlet temperature were measured by thermocouples in the cooling system of the HHF experimental platform. The minor-defects PFUs of the cooling water temperature difference during the cooling phase is the outlet temperature minus the inlet temperature (T(t) = T 2 −T 1 ).
The water temperature difference variation curve of minordefects PFUs under different loading and cooling modes were fitted separately by using equation (13). The surface temperature evolution of the minor-defects at different loading modes with the heat flux of 7.3 MW m −2 are shown in figure 8. It is reasonable to observe that, in Mode II and Mode III, temperatures are gradually increasing and then reaches an equilibrium state, the average temperature is 700 • C, then after stopping the loading of the heat flux, it starts to decrease and reaches an equilibrium state.
The major-defects PFUs was operated at the heat flux of 6.3 MW m −2 in Mode II (100 s ON/20 s OFF) and Mode III (400 s ON/20 s OFF). The cooling water conditions were set in the same conditions as for no-defects PFUs. The outlet temperature and the inlet temperature were measured by thermocouples in the cooling system of the HHF experimental platform. The major-defects PFUs of the cooling water temperature difference during the cooling phase is the outlet temperature minus the inlet temperature (T(t) = T 2 −T 1 ). The evolution curves of the cooling water temperature difference of the major-defects PFUs with time under different loading and cooling modes were fitted separately by using equation (13). Figure 7 illustrates evolution of the cooling water temperature difference with time for major-defects PFUs under the two modes and its fitted curve during the cooling phase. The results of the Mode II fit are shown in figure 9(a) with R 2 of 0.9215, the thermal decay time constant is 7.25 s with a standard error of ± 0.48 s. The results of the Mode III fit are shown in figure 9(b), with R 2 of 0. 94 256, the thermal decay time constant is 7.2 s with a standard error of ± 0.41 s. From the fitting calculations, the cooling water thermal decay time constant of major-defects PFUs does not vary with increasing loading and cooling modes time within the margin of error. Hence is set to a mean value of 7.22 s.
The surface temperature evolution of the minor-defects at different loading modes with the heat flux of 6.3 MW m −2 are shown in figure 10. It is reasonable to observe that, in Mode I and Mode II, temperatures are gradually increasing and then reaches an equilibrium state, the average temperature is 700 • C, then after stopping the loading of the heat flux, it starts to decrease and reaches an equilibrium state.

Flat-type mock-up.
The flat-type 2 mm W/Cu/CuCrZr mock-up was performed to alternate 15 s heating and 15 s cooling at heat flux of 20 MW m −2 . The inflow conditions of the cooling water loop were set as follows: inlet pressure: 1 MPa, inlet flow rate: 5 m 3 h −1 , inlet temperature: 20 • C. Its cooling water temperature difference during the cooling phase is the outlet temperature minus the inlet temperature (T(t) = T 2 −T 1 ). The evolution curves of the cooling water    HHF tests were performed on the flat-type 5 mm W/Cu/CuCrZr mock-up, loaded with 15 s heating and 15 s cooling at heat flux of 20 MW m −2 . The cooling water conditions are set in the same conditions as for the flat-type 2 mm W/Cu/CuCrZr mock-up. Its cooling water temperature difference during the cooling phase is the outlet temperature minus the inlet temperature (T(t) = T 2 −T 1 ). The evolution curves of the cooling water temperature difference with time were fitted by using equation (13). The experimental and fitted curves are shown in figure 11(b) with a good fit and R 2 of 0.992, the experimental curve is displayed as black solid lines and the fitted curves is displayed as red dashed lines. The thermal decay time constant is 1.77 s with a standard error of ±0.02 s.
HHF tests were performed on the flat-type 5 mm KW/Cu/CuCrZr mock-up, loaded with 15 s heating and 15 s cooling at heat flux of 20 MW m −2 . The cooling water conditions are set in the same conditions as for the flat-type 2 mm W/Cu/CuCrZr mock-up. The experimental data for the outlet temperature T 2 were measured by thermocouples in the cooling system of the HHF facility. The difference in water temperature is obtained by subtracting the inlet temperature (T 1 ) from the outlet temperature (T 2 ) (T(t) = T 2 −T 1 ). The evolution curves of the cooling water temperature difference for the cooling phase of the flat-type 5 mm KW/Cu/CuCrZr mock-up in Mode I was fitted using equation (13). The experimental and fitted curves are shown in figure 11(c) with a good fit and R 2 of 0.988, the experimental curves are shown as black solid lines and the fitted curves are shown as red dashed line. The thermal decay time constant is 2.01 s with a standard error of ±0.03 s.
The original experimental data and fitted data are shown in table 4. It is worth noting that the water temperature has reached thermal equilibrium after 6 s. As the number of experiments is too large to focus on the data before equilibrium, the water temperature for the first 6 s is screened according to a sampling frequency of 1:10, as is the data for the other experimental components.
The surface temperature evolution of the different flattype mock-up are shown in figure 12. The average temperature of the flat-type 2 mm W/Cu/CuCrZr mock-up, the flat-type 5 mm W/Cu/CuCrZr mock-up and the flat-type 5 mm KW/Cu/CuCrZr mock-up is 814 • C,1163 • C and 1160 • C respectively. The results indicate that the surface temperatures of three flat-type mock-up can quickly reach a steady state once the incident heat flux is exerted and can be quickly cooled down to a steady state by the cooling loop once the incident heat flux is stopped in loading cycle. Figure 13 shows comparison of the cooling water temperature difference with time for different models and its fitted curve during the cooling phase. Figure 13(a) represents results of the different flat-type mock-up, figure 13(b) represents results of the different defective mono-block model. Table 5 gives the thermal decay times obtained by fitting the characteristic curves of the cooling phase of the water temperature with time for different configurations of the bias filter components. It is reasonable to observe that the thermal decay time constants are different for different configurations of the divertor, indicating that the time constants are a unique property of the divertor. The experiments demonstrate that the time constant remains constant within the error range for the experimental part under different heat flows and for different loading times under the same heat flux. The value of thermal time constants for the mono-block are larger than those for the flattype mock-up, which shows that the heat transfer performance of the flat-type structure is better than that of the mono-block mock-up, in line with the heat flux that the component can withstand, which is 20 MW m −2 for the mono-block mock-up and 10 MW m −2 for the mono-block mock-up.  The thermal decay time constant for no-defects PFUs is smaller than those for PFUs with minor and major defects, indicating that no-defects PFUs have better heat removal capability than those with minor and major defects. For monoblock model, the larger the defect, the larger the thermal decay time constant, the lower the heat flux that can be tolerated and the poorer the heat transfer capability. If changes in water temperature can be monitored in real time, it is possible to determine the degradation of components over the course of their service.

Results and discussion
The time constant values for different thicknesses of PFC structures are different under the same connection process and the same material. The flat-type 2 mm W/Cu/CuCrZr mock-up has a smaller time constant value, indicating that the heat transfer performance of the flat-type 2 mm W/Cu/CuCrZr mock-up is slightly better than the flat-type 5 mm W/Cu/CuCrZr mockup. The time constant for different PFC materials are different under the same connection process and the same size. The time constant value for the flat-type 5 mm W/Cu/CuCrZr mock-up is smaller than the flat-type 5 mm KW/Cu/CuCrZr mock-up, Figure 13. Comparison of the cooling water temperature difference with time for different models and its fitted curve during the cooling phase: (a) flat-type, mock-up (b) mono-block, mock-up. indicating that the flat-type 5 mm W/Cu/CuCrZr mock-up has better heat transfer performance than 5 mm KW/Cu/CuCrZr mock-up. The reason for the difference is the different physical parameters of the respective PFCs.

Conclusion
A one-dimensional heat transfer analysis was used to establish a model for the thermal decay time constant, which led to the development of a method for obtaining the thermal decay time constant by fitting a temperature response curve using cooling water temperature data from HHF tests. The thermal decay time constant model theoretically demonstrates that the thermal decay time constant is an inherent characteristic of the system and depends only on the geometry and material properties, independent of the external loading and cooling conditions. The structure of the model also shows that the smaller the decay time constant, the faster the system response and the better the heat transfer performance.
The results of HHF tests on the same component under different loading and cooling conditions were analyzed using the water temperature data fitting method. The fitted thermal decay time constants remain constant, experimentally verifying the property that the thermal decay time constants are independent of the external conditions within error. With this method, the results of HHF tests on different components are analyzed to obtain the thermal decay time constants for HHF components of different constructions. The thermal decay time constants of PFCs with different thicknesses of PFMs vary considerably under the same material, processes and construction conditions.
The lower thermal decay time constant for a tungsten layer thickness of 2 mm indicates that the heat transfer performance is better than that of a component with a tungsten thickness of 5 mm. The thermal decay time constants of different PFMs are also significantly different under the same process, structure and geometry.
For the same 5 mm thickness of PFM, the thermal decay time constant of the flat-type W/Cu/CuCrZr mock-up is smaller than that of the flat-type KW/Cu/CuCrZr mock-up, indicating that the heat transfer performance of the flat-type W/Cu/CuCrZr mock-up is less than that of the flat-type KW/Cu/CuCrZr mock-up.
Under the same conditions of material, process, structure and geometry, the thermal decay time constants are also significantly different for no defects, minor defects and major defects. When defects occur within a component, its heat transfer capacity decreases and the value of the thermal time constant becomes larger, so it can be used to monitor the degradation of the object's performance.