Corresponding author: Aleksandra V. Voronina (AVVoronina@mephi.ru)

Academic editor: Yury Korovin

A procedure has been developed to determine the geometrical parameters of fuel assemblies (

The paper presents a mathematical model of the acoustic path developed in a geometrical acoustics approximation and its verification results. The model was used for computational and experimental studies of the ultrasonic test technique, and engineering formulas have been developed to calculate the errors of the transducer-measured distance to the

The developed technique to determine the VVER-1000

In the process of irradiation in the reactor core, a fuel assembly (

The

The inspection and repair bench developed for the NPP 2006 project uses contact differential transformer linear displacement transducers (

Despite the extensive use of ultrasonic techniques in technology, there are no currently common tools for developing ultrasonic

The paper describes a new ultrasonic pulse-echo technique to identify the VVER

A method has been developed for the _{б} (Fig.

Arrangement of ultrasonic transducers for the

The near-region dimension, _{б}, is equal to

_{б} = ^{2}∙

where

For example, for a transducer with a radius of _{б} = 0.33 m.

With the investigated surface being at a distance not exceeding 0.7 _{б}, the amplitude of the reflected _{б} will make it possible to stabilize the metrological characteristics of the ultrasonic technique.

The system with ultrasonic transducers is moved along the

where τ is the time of the

The pairwise arrangement of the transducers allows determining the face rotation angle, α:

α = arctg(Δ_{n}_{n}

where _{n}^{th} face, m; and _{n}^{th} face, m.

The computed angle permits an allowance to be made in determining distance

^{–1}+

where

To employ the angular allowance, it is required to determine initially the distance from the transducers to the surface based on formula (2), determine further the face rotation angle, α, and the distance with regard for the allowance using formula (4), and then calculate again angle α. This approach used in measurements will make it possible to reduce the influence of the angular incidence on the technique results.

The torsion angle, ψ, for each

ψ = ψ_{0} – ψ_{i}, (5)

where ψ_{0} is the upper or lower _{i} is the ^{th}

The deflection is calculated from the array of the

This method can be used for all types of the light-water reactor FAs, including square-shaped FAs.

A boundary layer occurs during the inspection of an irradiated

Since ultrasonic speed depends on the wave propagation medium temperature, an error takes place in determining the distance between the transducer and the

To determine the dependence of the distance measurement error on the factors described herein, a mathematical model of the ultrasonic technique acoustic path has been developed (

We shall introduce the following notation: τ_{1} (time for the _{2} (time for the _{3} (time for the reflected _{A} (time defined by the pulse-echo leading-edge recording technique); _{t}_{∞}) (constant sound speed value at the _{∞} far from the _{0} and maximum signal amplitude, _{max}).

The

(7)

The first three summands are determined from the _{0}, from which time τ is measured. When _{0} increases, Δτ_{A}

According to (7), calculating the

Based on semi-empirical relations known in literature to determine the heat-exchange characteristics during natural convection, a program was developed for calculating the speed of sound in water (

In the process of justifying the applicability of the developed acoustic path model, its verification procedure was undertaken (

Fig. ^{2} with a constant distance from the transducer to the inspected surface. The calculation and experiment results agree well across the range of Rayleigh numbers, specifically for a turbulent mode with Ra > 1∙10^{14}.

Measurement error |Δ

The influence of the slope on the ultrasonic measurement results is shown in Fig.

Distance increment measurement error,

It can be seen that the error is the larger the larger is the slope. The experimental data numbered 1 (discs) are above the reference line, but the dependences of errors are identical. The data numbered 2 (squares) demonstrate the results of employing the angular allowance when calculating the distance between the transducer and the surface using formula (4). When the proposed allowance is used, the error is smaller and does not exceed 5 μm in a range of 0 to 3°. It can be concluded based on this that the given allowance needs to be used for the distance calculation since it makes it possible to reduce considerably the measurement error.

Investigating the influence of natural convection on the ultrasonic dimensional measurement results has shown that the error of measuring the distance from the transducer to the inspected item for a turbulent convection mode is several times as large as for a laminar mode and is defined primarily by the boundary layer thickness and water temperature drop. For a turbulent natural convection mode that starts at Ra > 4∙10^{13}, nomograms were obtained for determining the boundary layer thickness, the temperature drop, the origin of the turbulent mode coordinate near the

Fig. ^{2} with an increment of 0.5 kW/m^{2}. The ordinate axis shows the absolute error value, that is, the measured distance is smaller when the heat flux density and the

A nomogram to determine the absolute error value obtained as a result of nuclear simulation; the experimental data for values ^{2} have been taken from (

Fig. ^{2}. A comparison of the calculated and experimental data shows these to agree satisfactorily.

The nomograms were processed to obtain engineering formulas to calculate the measured distance error for a turbulent convection mode (

_{т} = ω∙^{−0.25} (9)

where |Δ_{т} is the turbulent mode onset coordinate, m; ^{2}; Ra_{y}^{4}/(λν^{2}) is the local Rayleigh number; Pr is the Prandtl number; g is the free fall acceleration, m/s^{2}; β is the liquid bulk thermal expansion factor, 1/K; λ is the liquid conductivity factor, W/(m∙°С); and ν is the liquid kinematic viscosity factor, m^{2}/s.

The factors in formulas (8), (9) are determined depending on the

The decay heat value for the random burn-up and

A code has been developed based on the acoustic path mathematical model to simulate the VVER

The developed mathematical model and the code were used to simulate computationally and experimentally the VVER-1000

Simulation of the VVER-1000

The diagram presents dependences of the transducer-measured distance error, Δ^{16}, the water temperature being equal to 20 °C.

The dependence of the error on distance

The developed technique has been introduced in the TVSA-T

The process of the

A noncontact ultrasonic pulse-echo technique has been proposed to measure the VVER-1000 FA transverse dimensions, deflection and torsion angle in the NPP cooling pool conditions. Specific to the technique is simplicity and versatility in terms of its applicability for monitoring different FA designs, including TVSA and TVS-2M with any number of spacer grids.

A mathematical model of the acoustic path has been developed and verified in a geometrical acoustics approximation, taking into account the ultrasonic wave angular incidence onto the FA surface and the UW propagation in the natural convection conditions along the FA surface.

A code was developed based on the mathematical model and engineering formulas were obtained which make it possible to simulate the VVER FA form change monitoring, and design new monitoring systems based on ultrasonic pulse-echo technique for measuring linear dimensions.

The developed technique was used to build equipment for the VVER-1000 TVSA FA form change monitoring at units 1 and 2 of the Temelin NPP, the Czech Republic. The results of measuring ~ 40 assemblies with a burn-up of 11 to 52 MW·day/kgU and the cooling time of several days to two years after the withdrawal from the reactor core have proved the technique to be reliable and efficient.

^{th}Conference on Reactor Materials Science. Dimitrovgrad, 5–8. [in Russian]

* Russian text published: Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2022, n. 1, pp. 66–78.