Research Article 
Corresponding author: Mikhail T. Slepov ( slepovmt@nvnpp1.rosenergoatom.ru ) Academic editor: Yury Korovin
© 2022 Gennady V. Arkadov, Vladimir I. Pavelko, Vladimir P. Povarov, Mikhail T. Slepov .
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Arkadov GV, Pavelko VI, Povarov VP, Slepov MT (2022) Phenomenology of acoustic standing waves as applied to the VVER1200 reactor plant. Nuclear Energy and Technology 8(1): 3742. https://doi.org/10.3897/nucet.8.82755

The insufficiently studied issues of acoustic standing waves (ASW) in the main circulation circuits of the VVER reactor plants are considered. For a long time no proper attention has been given to this phenomenon both by the researchers and NPP experts. In general, generation of ASWs requires the acoustic inhomogeneities of the medium in the planes perpendicular to the direction of propagation of the longitudinal wave, in which a jump in acoustic resistance occurs, this is shown by the authors based on an example of the wave equation solution (D’Alembert equation) for a certain function of two variables. The ASW classification has been developed based on the obtained experimental material, 6 ASW types have been described, and their key parameters have been specified. The amplitude distributions have been plotted for all major ASW types proceeding from the phase relations of signals from the pressure pulsation detectors and accelerometers installed on the MCC pipelines. The nature of these distributions is general and they are valid for all VVER types. For the first time the globality of all lowest ASW types is identified. Four attribute properties of the ASWs have been formulated. The first attribute is the regular ASW temperature dependences, which is the source of the diagnostic information in the process of heating/cooling of the VVER unit. The linear experimental dependences of the ASW frequencies on coolant temperature have been obtained. The frequencies, at which the MCC resonant excitation due to coincidence of the ASW frequencies with the RCP rotational frequency harmonics, have been found experimentally. The ASW energy, which origin has resulted from the RCP operation, is estimated. The RCP operation can be presented as continuous generation of pressure pulsations, which fall onto the acoustic path inhomogeneities in the form of a traveling wave and generate a standing wave after reflection from them.
Acoustic standing wave, VVER1200, pressure pulsation detector, reactor coolant pump, main circulation circuit, autospectral power density, crossspectral power density, core, technical diagnostics
No proper attention has been given to vibroacoustic studies in reactor engineering. Among many factors of equipment wearing, the vibration is not considered to be the main factor of life expiration in the reactor plant designs and for vibration there are no limiting levels specified by Chief RP Designer, both for frequency and amplitude values. The acoustics of circulation circuits as such is not also taken into account when the reactor plant safety is justified in any Russian or foreign design (
In general, the wave equation (D’Alembert equation) for the certain function of two variables U (x, t) has the form
$\frac{{\partial}^{2}U(x,t)}{\partial {t}^{2}}={C}^{2}\xb7\frac{{\partial}^{2}U(x,t)}{\partial {x}^{2}}+f(x,t)$. (1)
The equation is presented in an elementary form for a onedimensional spatial coordinate and time function U (x, t). For a multidimensional and, in particular, threedimensional space, (x, y, z), the equation is written using the Laplace operator:
$\frac{{\partial}^{2}U(x,y,z,t)}{\partial {t}^{2}}={C}^{2}\xb7\Delta U(x,y,z,t)+f(x,y,z,t)$, (2)
where D = ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} + ∂^{2}/∂z^{2} is the Laplace operator; f (x, y, z, t) is the external forcing impact; and C is the longitudinal wave propagation speed in the medium.
A homogeneous (with no external forcing) wave equation can describe acoustic standing waves (ASW) originating from the certain initial stationary action U_{0}(x). The solution is a complex harmonic function of time, e^{j}^{wt}, with a spatially dependent amplitude, U_{0}(x):
U (x, t) = U_{0}(x)×e ^{j}^{wt}. (3)
With a sinusoidal spatial excitation, a standing wave has the properties of
The fundamental properties of the complex harmonic wave presentation and connection of the Helmholtz and D’Alembert equations with the ordinary differential Lagrange equation are described in detail in (
The new VVER1200 reactor facility, while differing from the VVER1000 reactor in terms of weight and dimensions, has other parameters of the structure and coolant natural oscillations. The risk of the natural and induced oscillation frequencies to coincide or the socalled resonant excitation threatens with an increase of the oscillation amplitude which shortens the reactor life. The average coolant temperature in the VVER1200 core is higher than in the VVER1000 and the socalled subcooled coolant boiling is expected to shift the ASW frequency harder. Even a minor increase in the coolant’s vapor fraction is capable to reduce considerably the ASW frequency. Judging by this factor, one should expect the VVER1200 ASW frequencies to decrease as compared with the VVER1000. It is not practically possible to estimate by calculation the resultant effect of the change in the properties of the VVER1200 ASW, as compared with the VVER1000 ASW from the above factors. The variation in the parameters of any spectral characteristic peak depends on its origin. Where a natural oscillation resonance is observed, small (in terms of the absolute value) natural frequency variations need to be found. With natural oscillations, the resonance amplitude, frequency and quality are associated with the weight, rigidity and damping of some structural element. The natural frequency, as it changes following a small rigidity variation, may correspond to major changes in the properties of the structural supports. Already a minor natural frequency variation may signal the onset of the defect growth and is, therefore, an important diagnostic sign.
If the vibration resonance has been caused by external forcing, the change in its parameters may correspond to the change in the properties of both the structure and the external force. Here, the limits for the permissible resonance amplitude, frequency and quality changes are much greater than in the event of natural oscillations. The occurrence nature and specifics of the ASW as the key oscillation excitation source for the internals, the reactor vessel and the RCP had been explored for quite a long time (
The NVNPP experts have developed a methodology of measurements using a portable analyzer as a versatile integrator for interconnecting different measuring systems as a single measuring cluster. A 40channel instrument, LMS SCADAS Mobile, was used as the analyzer. This is a multipurpose mobile analyzer for measuring and analyzing signals of dynamic processes that is compatible practically with any detector type (accelerometers, bridge detectors, microphone transmitters, speed sensors, thermocouples). It is capable to operate with a PC or a laptop through the Ethernet interface, via a wireless interface or as a standalone recorder. The analyzer is controlled using the LMS Test.Xpress software that includes the accelerator programming, channel calibration, measurement parameter adjustment, measurement control and data analysis functions. More detailed information on measurements is provided in (
Thanks to a great deal of experimental work undertaken at different stages of the VVER1200 commissioning (
ASW
_{L1} (loop ASW 1) is the most lowfrequency local simple ASW covering the whole of the loop. ASW_{L1} has one node on the pit wall in the reactor vessel’s hot nozzle and the second node on the pit wall in the reactor vessel’s cold nozzle, and the antinode is on the Ushaped piping, approximately in the loop middle (see Fig.
The standing wave, following ASW_{L1} in terms of the frequency growth, has zero phases in any pair of signals from four PPDs in four hot pipelines (see Fig.
According to the known distribution of the ASW_{L1} amplitudes, it is not difficult to proceed to the distribution of the amplitudes of its harmonics (see Fig.
It is not difficult to plot the fourth harmonic for ASW_{L1} as well. We shall denote it by ASW_{L4} and shall refer to it as loop ASW 2 (see Fig.
Also evident are simple local ASWs which cover the MCC’s hot and cold pipelines with their halfwave (see Fig.
The ASW data as applied to the VVER1200 reactor plant is shown in Table
No.  ASW description  Designation  ASW frequency, Hz with tCL = 40 °C  ASW frequency, Hz with tCL = 290 °C  ASW halfwave length, m 

1  Loop ASW 1  ASWL1  11.59  7.09  57.6 
2  Vessel ASW 1  ASWV1  15.86  9.96  33.1 
3  Loop ASW 2  ASWL2  21.46  13.53  28.8 
4  Loop ASW 4  ASWV4  26.10  40.93  14.4 
5  Hot ASW  ASWH1  –  –  15.8 
Cold ASW  ASWC1  –  –  15.3 
The ASW central frequency varies as the coolant temperature varies (during the unit heatup or cooldown). A monotonous increase in the coolant temperature causes the ASW central frequency to move monotonously towards 0 Hz. Spectra of the pressure pulsations in the hot leg of the VVER1200 primary loop at temperatures of 121 до 286 °C are presented as an example in Fig.
The frequencies of all standing waves start moving and, importantly, each ASW central frequency has its own rate [Hz/deg.] of movement along the frequency axis. The process of the coolant heatup by ~ 300 °C leads to the coalescence of some reactor coolant pump (RCP) rotational frequency harmonic with the central frequency of some ASW (or its harmonic). Such calescence cannot be allowed in the nominal mode of operation with a 100% level of the reactor power but is inevitable in dynamic modes (
It was proved experimentally in the course of the studies that there is resonant excitation as the coolant temperature reaches ~ 170 °C (mode 2; ASW_{L2}) and ~ 190 °C (mode 3; ASW_{L4}). These resonant excitation modes have not been taken into account by the chief designer in estimating the endurance and are not described in operating documentation as critical modes requiring the minimum time for these to be over. This adds importance to vibration measurements in the process of commissioning when VVER units are put into operation. Practically valuable is the approach in accordance with which the vibration state of a new VVER1200 reactor is the same as for VVER1000 in the sense that the above frequencies vary just slightly (
The standing wave energy owes its origin to the RCP operation (
We shall estimate the energy spent by the RCP to generate and maintain the ASW in the MCC. The ASW specific energy in a volume unit is
E (x, t) = I (x, t)/c, J/m^{3}, (4)
where I (x, t) is the ASW power flux density, and Н/(m×s), s is the speed of sound.
By solving wave equations for the longitudinal wave pressure, P (x, t), and speed, V (x, t)
∂^{2}P (x, t)/∂x^{2} = c^{–2}×∂^{2}P (x, t)/∂t^{2}, (5)
∂^{2}V (x, t)/∂x^{2} = c^{–2}×∂^{2}V (x, t)/∂t^{2}, (6)
we shall get the following complex expressions
P (x, t) = A×e^{–j (kx – wt)}, V (x, t) = A×e^{–j (kx – wt)}, /(r×c), (7)
where A is the pressure wave amplitude; and r is the coolant density.
The ASW power flux density modulus is then
I (x, t) = 〈ReP×ReV〉 = A^{2}/(2×r×c^{2}), (8)
where 〈 〉 is the averaging for the period; and Re is the real part.
The ASW power for the whole of the RCP is estimated by the expression
N (x, t) = V_{k}×E (x, t)×w_{ASW}, (9)
where V_{k} is the RCP volume; and w_{ASW} is the ASW frequency.
The ASW power in terms of one circulation loop is
N _{1}(x, t) = 0.25V_{k}×E (x, t)×2p×f_{ASW} =
0.25(M_{CL} r)×0.5A^{2}/(r×c^{2})×2p×f_{ASW} =
= 0.25p×f_{ASW}×M_{CL}×[A/(r×c)]^{2}, (10)
where M_{CL} is the coolant mass in the MCC.
We shall define that the pressure pulsation value, А, is in a broad range of 1×10^{5} to 1×10^{6} Pa, according to (
Let us note that the RCP (
The above estimate is rough but it gives an idea that the existing RCP configuration and the field of different ASWs may lead to up to 9% of the RCP power lost.
The use of vibroacoustics in reactor engineering advances extremely slowly, possibly, due to the departmental barriers, due to the complexity of mathematical tools, science intensity of diagnostic systems, and absence of normative requirements regulating the acoustic and vibration effects in the MCC and their continuous monitoring (