Position estimation of fuel ball blockage in pipeline

When the high-temperature gas-cooled reactor is working, lots of fuel balls are transfered in the core pipelines. To enhance the operation efficiency, this paper proposes a simple and feasible estimation method for pipeline blockage detection. This method steps include as follows: Firstly, the three-dimensional models of possible blockage pipelines are formulated; once the blockage breaks out, the radiation counting rates are collected along a predefined route; subsequently, the estimation space is set from above models, and the particle filter algorithm is executed to complete the processes of weight updating, normalization and resampling; finally, the blockage position is calculated from the convergent particle set. Simulation experiments showed that this method could effectively work and accurately estimate the blockage location.


Introduction
Compared with traditional nuclear power reactors, high temperature gas-cooled reactors have the characteristics such as high-power generation efficiency, good safety, continuous operation and small modular construction.It has the potential to become one of the priority development reactors for the fourth-generation nuclear energy system.In normal operation, there are lots of fuel balls in the core of high temperature gas-cooled reactor.Due to the huge number of radioactive fuel balls, blockage may occur at special positions such as pipeline bending or pipeline passing through walls during transmission.Once blockage occurs, the power plant needs to shut down for refueling, which brings potential safety hazards and huge economic losses.
In order to monitor the fuel ball movement in reactor pipelines, predecessors have carried out lots of studies.The mainstream method is to detect the induced electromotive force generated from the passing ball.When the ball passes through one special detection coil, it will produce induced electromotive force, and its passing situation can be determined from the waveform.But this type method requires placing the coil into the pipeline, which can damage the high-pressure airtightness of the pipeline and make maintenance more troublesome [1,2].In order to deal with the monitoring and detection of radioactive fuel, Liu designed and built a two-dimensional experimental bench.Through this platform, his team deploys excitation coils and detection coils outside the pipeline to check the induced voltage from balls [3,4]; Wan invented a device like a CT machine, which can scan the pipeline and found out the blockage position [5].However, their methods face some problems, such as long detection time and high costs.
To overcome these problems, this paper proposes a simple pipeline detection method that does not require fixing the detection position.The method involves establishing accurate three-dimensional geometric models for the transmission pipelines, collecting radiation counting rates in the possible jam area using a radiation detector, defining the estimation space of possible blockage positions, and executing the particle filter algorithm to update weights, normalize and resample the filtered particles, and identify the blockage position of the radioactive source.The effectiveness of the method was verified through targeted simulation, which showed that it could quickly and accurately identify the blockage area.

Pipeline model outside the wall
The physical model of the transmission pipeline is established in a certain simulation software to simulate the through-wall straight pipeline and curved pipeline that are prone to cause blockage in the reactor core.Here, the bend diameter is 1600mm, the maximum distance between each group of bend pipelines is 600mm, and the narrowest point is 200mm.The straight pipeline is an extension of curved pipeline, and the blockage probability is less than that of curved pipeline.Below are the pipelines models for the later simulation experiment.
(1) Linear pipeline model The mathematical expression of four straight pipelines is as follows: (5.3, 5.55) 7.9 34.98 z 8 (5.3, 5.55) 7.9 44.98 z 8 x (2) Arc pipeline model Four curved pipelines can be considered as four semicircles, the circular plane includes these semicircles is the intersection of the sphere where the circle is located, and the plane where the circular plane is located in space.The expression is as follows, where (x 0 , y 0 , z 0 ) is the center of the circle: The four curved pipelines are all semicircular with a radius of 0.8m, and the specific parameters are as follows: Table 1 Figure 1a provides a visual representation of the pipeline and its location within a concrete wall.This is important for the simulation as it allows for testing the effect of the wall's shielding on the particle behavior.Figure 1b illustrates the position of the radioactive source (point O) and the observation point (point A) with a schematic diagram.It provides a clear understanding of the relative positions of the source and observation point, which is crucial for the estimation method proposed in this paper.

Nuclear radiation measurement
If there is an isotropic γ-ray point source in vacuum, the activity A S of which emits only one photon per decay and the photon emissivity is 100%.For the observation point A, the counting rate I A at it can be approximately expressed as [6]: The above formula I A represents the counting rate (cps) at point A, A S represents the radioactivity (Bq) of the radioactive source, R A represents the distance (m) from the radioactive source to point A, ρ represents the detection efficiency of the radiation detector, and S represents the effective detection area (m 2 ) of the radiation detector.The attenuation of γ-ray radiation by air is weak and negligible, but it is obvious when there are shields such as lead, iron and concrete.Considering the background counting rate n b (cps) caused by other factors in the environment except radioactive sources, after measuring in τ seconds, the radiometer value λ A at point A is as follows [7]: Where μ m is the linear attenuation coefficient of the m-th shield; △ m is the thickness (m) of the mth shield.In fact, it is usually difficult to obtain parameters such as effective detection area and detection efficiency.If the count rate measured at a distance of 1 m from the radioactive source I s represents the "intensity" of source, λ A can be approximated as: In this paper, the medium is only concrete wall and air, and the value of m is 2; For concrete wall, let μ 1 be its attenuation coefficient and take the value of 0.05; for air, let μ 2 be its attenuation coefficient and take the value of 0; The background counting rate is low in the environment, assuming that the value of n b is 10cps and the measurement time τ is 10s.Then, the formula (8) can be expressed as: The relationship between the theoretical measurement value of radiation detector and various factors is introduced above.The research shows that the relationship between the theoretical measurement value and the actual measurement value obeys Poisson distribution law [8].If the theoretical counting rate per second at a certain place is μ, the actual counting value (c ) P ;λ in τ seconds at that place is as follows: In the above formula, P represents Poisson distribution, and λ=μτ represents the mean value of Poisson distribution.

Position estimation of blockage in pipeline
The basic idea of particle filter algorithm is to use a group of discrete particles to approximate the posterior probability density function of system state.When determining the blocked ball position in pipeline, the particles are filtered mainly through observation values, so that the particles converge near the radioactive ball in constant change.The main steps include particle initialization, weight update and normalization, particle resampling and so on.
(1) Particle initialization N particles with respective weights ω 0 are generated by using uniform distribution in the threedimensional search area.The parameter of each particle is ] T , which contains the position of the radioactive source (x 0 i , y 0 i , z 0 i ) and the activity of the radioactive source I 0 i .During initialization, all particles constitute a particle set｛X 0 i , ω 0 i ｝ N i=1 , and the weights of each particle produced by uniform distribution are equal, that is, ω 0 i =1/N.
(2) Weight updating and normalization Given the parameter X t i = [x t i , y t i , z t i , I t i ] in the particle set｛X t i , ω t i ｝ N i=1 , the corresponding theoretical count value λ t ( X i t ) for each particle can be obtained by equation (9).Then, according to the t-th observation value C t obtained by detector, the corresponding weights of each particle can be obtained from the Poisson distribution probability of equation (10), and ω i t in the particle set is updated as: When the blockage position is inside the wall, the schematic diagram is as Figure 1a follows.
For the fuel ball blocked inside the wall, it is also necessary to collect the radiation dose information and position coordinates of multiple observation points, and the blockage position is estimated with particle filter.Finally, the weight value is normalized,  is the normalized importance weight value, and the expression is as follows: (3) Particle resampling Before particle resampling, the degradation degree of particles is judged by the effective sample number N eff , and then whether to resample is decided.The approximate value of N eff is as follows:   The threshold value of N eff is set to 2N/3.When N eff < 2N/3, the variance of particle set weight is large, and the resampling step is performed at this time, otherwise the particle set remains unchanged.

Simulation experiments
To verify the effectiveness of this method, simulation experiments are carried out.Since the actual pipeline scene may involve both shielded and unshielded cases, simulations are conducted separately for each case.In the unshielded case, the orbital scattered particles are directly processed by the particle filter algorithm.However, in the shielded case, the method needs to take into account the type of concrete material and the depth of the wall to properly account for the shielding effect.Therefore, simulations are carried out accordingly for both shielded and unshielded cases to accurately evaluate the performance of the proposed pipeline detection method.gradually converge to the position where the ball is blocked.In the simulation, the blockage position is set on a straight pipeline with coordinates (5.28m, 3.25m, 8m).After 11 observations, the particles are concentrated near the position of the blockage, and the estimated position of radioactive source is (5.27m, 2.99m, 8m) by weighted summation.The simulation diagram and data are as Figure 2b and Table 2b shows.Table 2b shows the position coordinates of the 11 observation points and the estimated position corresponding to each time.The coordinates of radioactive sources set as (5.28m, 3.25m, 8m), and the final estimation values are (5.27m,2.99m, 8m) with an error of 0.26m.
The method used in this simulation, such as the number and distribution of particles, the range and direction of observation points, and the linear attenuation coefficient of the wall, should also be adjusted according to the specific application scenario to achieve better results.

Conclusion
This paper proposes an estimation method based on particle filter to find the blockage position in a reactor pipeline.By collecting radiation count rate values along an observation path, the method estimates the position of the blockage.Compared to traditional methods, this approach is more costeffective and avoids damaging pipeline integrity.However, as the observation point is located far away from the pipeline, the accuracy of the estimation may need to be improved in future work.Overall, this method shows promising potential for practical application and further optimization.

.Figure 1 .
Figure 1.Schematic diagram of pipeline passing through the wall (a) Estimation of radioactive sources blocked in walls(b) Schematic diagram of wall shielding.