Corresponding author: Svetlana A. Kachur ( kachur_62@mail.ru ) Academic editor: Yury Kazansky
© 2019 Svetlana A. Kachur.
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:
Kachur SA (2019) Diagnostics of the critical heat flux state of a VVER reactor based on a channel steaming model. Nuclear Energy and Technology 5(2): 139144. https://doi.org/10.3897/nucet.5.36475

The purpose of the study is to develop a model for predicting the process of a critical heat flux state with the VVER reactor core channel steaming. The model describes the dynamics of the nuclear reactor behavior in conditions of uncertainty, which are typical of abnormal situations, based on information on the process of heat exchange in the core process channels.
The use of the proposed model leads to an increase in the speed of response due to a simplified procedure to calculate the parameters of the heat exchange process in the reactor core. The quality of the reactor state assessment is improved through the prediction of the heat exchange process parameters and determination of the critical heat flux parameters in the core prior to the onset of surface boiling the potentiality of which is not predicted in modern VVER incore monitoring systems.
A modification of the mathematical model has been proposed which offers the simplest possible way of using the advantages of neural networks in diagnostics. The model can be used to develop systems for diagnostics of incore anomalies and systems for adaptive control of the VVERtype reactor thermal power.
Nuclear reactor, critical heat flux, power density, thermophysical model, identification, neural networks
The cores of large NPP reactors have complex structures with many fuel assemblies and control rods operating in stress conditions. Solving the problem of optimizing the power density within such cores and improving the cost effectiveness of the NPP operation required the development of dedicated tools and automatic devices for monitoring and control of nuclear reactors (
In a general case, the required scram signals and settings are based on computational and experimental studies and depend on the reactor type. Often, the parameters that define the reactor plant safety (fuel and cladding temperature, hot spot location, boiling point, etc.) cannot be measured directly. In this case, safe variation intervals of measured parameters and the core settings for each of them need to be determined based on physical and thermal engineering calculations.
Describing the operation of a nuclear power plant (NPP) during local disturbances as a random process, the key indicators of which in the maximum thermal loading mode are the surface boiling parameters, namely the coolant pressure, the coolant temperature in the mixing chamber, and the average volumetric steam quality (
On the one hand, there is a need for building simple and effective models of the heat exchange process in the reactor fuel channel which allows one to conclude, based on a small number of parameters, that the heat exchange is abnormal at early stages. On the other hand, this requires the development of fastresponse adaptive control systems the models of which would take into account the stochastic nature of the heat exchange process.
The mechanisms of a critical heat flux in channels depend to a great extent on the twophase mixture flow mode, the liquid subcooling and the heat flux density (
Many papers (
Recently, there has been a heightened interest in using the capabilities of neural networks for the NPP monitoring and control (
Development of a mathematical model for the heat generation in a nuclear reactor includes
Using the results of analyzing models of critical phenomena during boiling in a twophase flow, an empirical model of determining the steam quality for different boiling stages, based on Z.L. Miropolsky’s data, and empirical studies of surface boiling in a channel of the IR100 nuclear research reactor have been chosen to investigate the process of heat exchange in the reactor core (Fig.
The reactor model can be presented as an integrated model of particular FAs.
We shall assume that it is enough to know the following parameters of the current process in each channel to diagnose and predict the state of the heat exchange process in the nuclear reactor core:
As the model base, we shall select the Тsdiagram (
Stage 1 in the model development is to analyze the relationship between the system’s specific entropy s and the experimental acoustic spectral characteristics. It can be seen from Fig.
B = Δq⸱k_{norm}/ΔA = Δq/ΔT, (1)
where ΔA is the variation of the acoustic noise fluctuation spectral density amplitude in response to the specific heat flux density variation Δq; ΔT is the channel outlet coolant temperature variation in response to the variation Δq; and k_{norm} is the normalization factor depending on the structural features of the particular reactor.
After simple transformations, the following relations can be obtained from the heat balance equation with regard for (1) and with the assumption that ΔG/G = Δα/(1 – α) (
Δq = C⸱G⸱ΔT → Δq/ΔT = C⸱G ≡ B,
Δq = ΔG⸱C⸱T → Δq/T = ΔG⸱C =
= (ΔG/G)⸱G⸱C = Δα/(1 – α)⸱B, (2)
where q is the specific heat flux density; Т is the channel outlet coolant temperature; G is the coolant mass flow rate; С is the specific heat capacity; α is the volumetric steam quality; and Δa is the steam quality variation leading to the coolant mass flow rate variation ΔG.
The specific entropy variation Δs is determined with regard for (2) as follows:
Δs = Δq/T = Δα/(1 – α)⸱B. (3)
It follows from relation (3) that the maximum specific entropy variation is achieved provided that Δα/(1 – α) =1 and has a value of Δs_{max} = B.
Stage II is to determine the surface boiling onset temperature Т_{bo}. By analyzing Fig.
T _{bo} = T with θ = T_{s}. (4)
Stage III in the model development is to plot the work line in the Тsdiagram. The work line is defined by the point of the boiling onset (point А, Fig.
The work line equation has the form
T (s) = (s – s_{bo})(T_{be} – T_{bo})/(s_{be} – s_{bo}) + T_{bo}. (5)
Stage IV is to predict the steam quality a during surface boiling (Т_{bo} < Т < Т_{s}) in the event of the coolant temperature variation by ΔТ. Using line equation (5) for Т = Т_{bo}+ ΔТ, we determine Δs = s – s_{bo}. After simple transformations (4) and (5), assuming that Δα= α – α_{bo}, the value a is calculated by the formula
α = (α_{bo} + Δs/B) / (1 + Δs/B). (6)
The value α_{bo} is determined in accordance with the formula
α_{bo} = 1.17q^{0.35} / P^{0.15}(ρw)^{0.15}, (7)
where ρw is the mass velocity (
Stage V is to predict the steam quality in the event of the coolant temperature variation by ΔT. Using the relations for the surface boiling region (х_{0} < х < 0, Т_{bo} < Т < Т_{s}) in accordance with (
α = α_{bo} (1 – x/x_{bo})^{1.35}, (8)
x _{bo} = –0.573q^{0.7}(P/(ρw))^{0.3}, (9)
following simple transformations, we get
x = x_{bo}(1 – α^{0.74}/α_{bo}). (10)
Stage VI is to predict the specific heat flux density q_{pred} during surface boiling (Т_{bo} < Т < Т_{s}) in the event of the coolant temperature variation by ΔТ. Based on the current information on the value q and Т on the work line in the Тsdiagram, we determine Δq = Δs (Т + ΔТ) and calculate the predicted value
q _{pred} = q + Δs (T + ΔT). (11)
The presented model development procedure suggests monotonous variation of parameters.
The changes in the work line position for the point Р, provided there are random external impacts capable to lead to the critical heat flux, are as follows:
The physical meaning of the work line in the Тsdiagram can be defined as follows. For gases, the heat supply process can be nearly isothermic if consisting of alternating isobaric heat supply processes with a subsequent adiabatic expansion in a small interval of pressures (Fig.
In accordance with the work line АD (Fig.
We shall take entropy as the system state parameter.
Assuming that the distribution of the indicator s within the classes is described by normal law with the mathematical expectation s_{bo} for class 1 and sʹ_{K} for class 2 and with an equal dispersion of σ^{2}, Fig.
The coordinates of the point C (s_{cr}, Т_{cr}) on the work line (see Fig.
s _{cr} = (s_{bo} + sʹ_{K})/2 = s_{bo} + B/4, (12)
Δs = В/4, q = q_{bo}, ΔТ = Т_{cr} – Т_{bo}, make it possible to calculate a_{cr}, х_{cr}, and q_{cr} using formulas (6), (10), and (11).
The proposed model makes it possible to simplify to a great extent the calculation of such parameter as steam quality by substituting the iterative algorithm for its calculation by a sequence of several formulas. And the initial work line is plotted with a sufficient time to the boiling onset and until the need arises for monitoring the surface boiling process parameters.
It was assumed in the process of the mathematical model development that surface boiling was already taking place since the fuel wall temperature had reached the coolant boiling point. Therefore, the boiling onset temperature corresponds to the channel outlet coolant temperature when surface boiling occurs. In (
The coolant flow is initially convective. The boiling onset temperature requires to be predicted and the work line equation built based on this prediction (5). The transition from the convective phase to surface boiling is possible in the event an accelerated power variation process is taking place.
The work line’s maximum slope angle b will be defined by the secondorder differences of the entropy with Δs = B:
ΔT/Δ^{2}s = ΔT^{2}/Δ^{2}q = tg β, (13)
where Δ^{2}q is the secondorder difference of q.
We shall assume that the Тsdiagram is described by the function F_{Ts}. Since the distance between the point of the work line intersection with the phase equilibrium curve of the Тsdiagram corresponds to B, then the boiling onset point can be determined from the relations
F_{Ts} (s_{bo} + B) – F_{Ts} (s_{bo}) = B⸱tg β, (14)
T _{bo} = F_{Ts} (s_{bo}). (15)
In the event F_{Ts} is given in a tabulated form, the boiling point coordinates are searched for by simple enumeration until condition (14) is fulfilled with the preset error. The rest of the parameters are determined in accordance with the proposed mathematical model.
A neural network with one perceptron is proposed to be used for the rapid identification of the channel state. The value s_{cr} calculated using formula (12) divides the work line into two portions (two subsets of points). By training the perceptron such that the values of the work line points before s_{cr} will correspond to the zero class, and those after s_{cr} will correspond to class 1, it is possible to classify the given vector of the values (s, T) as one of the two classes. Based on the classification results, a message is displayed on if the current channel state complies with the requirements.
The proposed mathematical model makes it possible to improve the operating safety of such a complex system as nuclear reactor by defining the boiling process as a principal manifestation of its operation. The model offers an opportunity to identify and predict in a timely manner an emergency caused by worsened heat removal from fuel thanks to using direct measurements of the heat exchange parameters, minimizing indirect calculations and employing empirical formulas. The model extends the class of the problems addressed, that is, makes it possible to proceed from the problem of identifying the nuclear reactor parameters and state to the problem of the critical heat flux prediction.
A possibility has been considered for rapid diagnostics of the channel state using neural network technologies.