Investigation of heat generation in a pebble bed during induction heating

. The use of fuel in the form of microfuel filling in nuclear reactors is a possible way to improve the safety and efficiency of nuclear power plants. In experimental studies, microfuel elements are simulated by stainless steel balls, and internal heat generation by high-frequency induction heating. The paper presents the results of a study of heat release in a pebble bed during induction heating, and features are revealed. The distribution function of the volumetric density of heat release along the height of the pebble bed is obtained.


Introduction
Increasing the efficiency of nuclear energy while simultaneously improving safety requires new approaches.One of the promising solutions to this problem is the use of alternative fuel in the form of spherical microfuels.Fuel assemblies with microfuels, on the one hand, have inherent internal safety, and will also improve energy efficiency.In such an assembly, microfuel elements are supposed to be placed between perforated covers in the form of free pebble bed [1].Fuel assemblies based on microfuel elements can probably find application, first of all, in low-power nuclear power plants, the interest in which is currently growing.
To use microfuel elements in fuel assemblies, systematic studies of hydrodynamics and heat transfer during the flow of a single-phase and two-phase coolant are required.In such studies, microfuel elements are replaced by stainless steel balls, and internal heat generation is simulated by high-frequency induction heating.Few studies like this have been done.

Experimental stand and test section 2.1 Hydraulic circuit
For experimental research, a new experimental stand was created in the work (Fig. 1), the stand includes a hydraulic circuit, a system for measuring, collecting and processing information, a test section with pebble bed, a high-frequency induction heating system to ensure heat generation in the pebble bed.The stand is designed for the following operating parameters: coolant temperature up to 100°С, coolant pressure up to 1.5 MPa, coolant flow rate (0.01-0.50) kg/s, induction heating power up to 20 kW.Distilled water and dielectric liquids can be used as a heat carrier.

Test section
The test section with bebble bed (Fig. 2) consists of two coaxially arranged polycarbonate tubes (3,4).The outer tube (3) is pulled together by fiberglass flanges (1,10) using four studs (6).The inner tube with a diameter of D = 51 mm consists of four parts between which there are gratings with round holes with a diameter of 1.5 mm (2,5,8).Balls with a diameter of 2.0 mm in the form of free pebble bed (7) are placed in the inner tube between gratings (5) and (8).Pebble bed height H = 100 mm, porosity ε = 0.39.The balls are made of AISI 420 steel.The taps for measuring pressure drop (p1, p2) are made of a capillary tube with a diameter of 3 mm.The area with pebble bed is placed in an induction coil made of copper tube, the length of the coil is twice the height of the pebble bed.The temperature of the liquid and balls in the test section was measured by chromel-alumel cable thermocouples.The thermocouples with a cable diameter of 0.3 mm were manufactured to minimize disturbances introduced into the flow.The layout of thermocouples and their coordinates are shown in fig. 3. and in table 1, respectively.Thermocouples were placed in seven sections along the height of the pebble bed and in three coordinates along the radius.Several thermocouples in a pebble bed were combined into measuring elements for further determination of the heat transfer coefficient.A drawing of such an element is shown in fig. 4. The thermocouple (2) was embedded in the wall of the ball (1), the thermocouple (3) was located in the liquid in the immediate vicinity of the ball surface.The thermocouples were rigidly connected by twisting (4).

Experimental data
The first part of the experiments was carried out without a coolant flow through the test section and was aimed at identifying the features of heat release in a pebble bed during induction heating.The experiments were carried out as follows: the test section was filled with water, then the recording of thermocouple readings began, after which the induction heating installation was turned on.
Heating continued up to temperatures safe for the design of the TS.The experiments were carried out at an induction heating power of 2, 5, 8, 11, and 14 kW.Temperature rise over time for all thermocouples was linear.
Further the primary data were processed and the temperature distributions were obtained along the radial and axial coordinates of the pebble bed.For all capacities the distributions have a similar form.On fig. 5 shows the temperature distribution along the radial coordinate in the section z = 50 mm at a heating power of 8 kW for three time moment.It can be seen that the temperature distribution along the pebble bed radius is almost constant.This proves the absence of electromagnetic shielding for heating the central part of the pebble bed.On fig.6 shows the temperature distribution along the axial coordinate at a heating power of 8 kW for three time moment.There is an uneven heating of the pebble bed.The maximum temperature is reached in the central part of the pebble bed, near the ends of the pebble bed the temperature is noticeably lower.This result, apparently, is associated with the inhomogeneity of the magnetic field of the coil.The second part of the experiments was carried out in a stationary mode with a coolant flow, at mass flow rates of 0.016 -0.12 kg/s and induction heating power of 3, 5, 7, 11, 16 kW.The temperature distribution along the radial coordinate at an induction heating power of 16 kW and a coolant flow rate of 0.084 kg/s shows on fig.
7. The figure shows, similarly to a series of experiments without a coolant flow, an almost constant temperature along the radius of the pebble bed.At the points of location of thermocouples embedded in the ball, there is a difference in the temperature of the wall of the ball and the liquid, this is an expected result, because pebble bed is heated from the inductor and transfers heat to the coolant   The temperature distribution of the coolant along the height of the pebble bed at a mass flow rate of 0.084 kg/s shows on fig.8.In contrast to heating without a coolant flow, a uniform temperature increase along the height of the pebble bed is observed here, and the uneven heat generation in the pebble bed is imperceptible.However, to calculate the local heat transfer coefficients, the nonuniformity of heat generation must be taken into account.
q vvolume density of heat generation, W/m 3 , c p.f , c p.sspecific heat capacities of liquid and balls at constant pressure, εporosity of pebble bed, in experiments ε = 0,39.For all the investigated heating modes, according to formula (1), the experimental data were processed and the distribution of the volumetric heat generation density over the height of the pebble bed was obtained (Fig. 9).The data is well approximated by the cosine function written relative to the center of the pebble bed: The results of calculation by formula ( 2) are shown by lines in Figs. 9.The deviation of the calculation according to formula (2) from the experimental points does not exceed 10% (Fig. 10), therefore, when performing numerical simulation of heat transfer in pebble bed, as well as when calculating local heat transfer coefficients, heat generation along the pebble bed axis must be determined in accordance with the function (2).Publications are known in the literature, where the method of induction heating is also used to heat the pebble bed, for example [2].In this paper the height of the test section with pebble bed was 670 mm.When comparing the distributions of heat generation along the axis in a dimensionless form with the data presented in this work, quite satisfactory agreement is noted, both qualitative and quantitative (Fig. 11).Therefore, the volumetric density of heat generation function over the pebble bed height, obtained in this work, is universal.

Analysis of experimental data
We will obtain the function of volumetric heat generation along the height of the pebble bed based on the data on the temperature distribution along the height of the backfill without a coolant flow (Fig. 6) .During the operation time of the induction heating unit, the readings of all thermocouples in the TS grow linearly with different slopes.Knowing the slope coefficient for each thermocouple, it is possible to determine the volume density of heat generation by the formula: 1,2 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 q v /q vmax z/H P = 2 kW P = 5 kW P = 8 kW P = 11 kW P = 14 kW data [2]

Conclusion
The heat generation in a pebble bed during induction heating has been investigated.The absence of electromagnetic shielding for heating the central part of the pebble bed was revealed.In the axial direction, a significant inhomogeneity of heat generation was obtained, apparently associated with the inhomogeneity of the magnetic field of the inductor.The distribution function of the volumetric density of heat generation over the height of the pebble bed is obtained, which satisfactorily describes both our own experimental data and the data of other authors.
The work was supported by the grant of the President of the Russian Federation MK-4552.2022.4.

Fig. 5 .
Fig. 5. Temperature distribution along the radial coordinate in the section z = 50 mm at a heating power of 8 kW.

Fig. 6 .
Fig. 6.Temperature distribution along the axial coordinate at a heating power of 8 kW. .

Fig. 7 .
Fig. 7. Temperature distribution along the radial coordinate at a heating power of 16 kW and a mass flow rate of 0.084 kg/s.

Fig. 9 .
Fig. 9. Distribution of the volumetric density of heat generation along the height of the pebble bed.

Fig. 8 .
Fig. 8. Temperature distribution along the axial coordinate at a heating power of 16 kW and a mass flow rate of 0.084 kg/s

Fig. 11 .
Fig. 11.Distribution of the volumetric density of heat generation along the height of the pebble bed.

Table 1 .
Thermocouple location coordinates in TS.
Fig.4.Element for measuring the temperature of the ball wall and liquid.