Corresponding author: Svetlana A. Kachur (kachur_62@mail.ru)

Academic editor: Yury Kazansky

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 in-core 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 in-core anomalies and systems for adaptive control of the VVER-type reactor thermal power.

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 fast-response 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 two-phase mixture flow mode, the liquid subcooling and the heat flux density (_{cr}(_{cr}) (_{cr} is the critical density of the heat flux, and _{cr} is the critical steam quality), this requiring _{cr} to be calculated from a heat balance equation which leads to additional errors. Besides, the problem of nonterminal adaptive control for the nuclear reactor thermal power based on a quadratic criterion suggests that there is a system state vector (core channel thermal power vector) which cannot be obtained based on existing thermophysical and thermodynamic models.

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

plotting a diagram of the reactor state based on thermodynamic characteristics, experimental dependences and current variations in the key parameters;

prediction of the parameter variations and calculation of the parameter critical values;

predictability of an abnormal heat exchange at the surface boiling onset stage and/or in the event of a sharp variation in the heat exchange process key parameters (Kachur 2009).

Using the results of analyzing models of critical phenomena during boiling in a two-phase 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 IR-100 nuclear research reactor have been chosen to investigate the process of heat exchange in the reactor core (Fig.

Experimental data describing the channel temperature mode (a) and the acoustic spectral characteristics (b)

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:

channel outlet coolant temperature Т;

primary circuit pressure Р;

channel top fuel wall temperature θ;

specific heat flux density q;

pre-boiling temperature characteristics (Fig. 1a);

acoustic spectral characteristics (Fig. 1b).

As the model base, we shall select the

^{6} W/m^{2}) and subject to respective normalization, all characteristics have one slope that defines the coefficient value

_{norm}/Δ

where Δ_{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 Δ

Δ

Δ

= (Δ

where

The specific entropy variation Δ

Δ

It follows from relation (3) that the maximum specific entropy variation is achieved provided that Δα/(1 – α) =1 and has a value of Δ_{max} =

_{bo}. By analyzing Fig. _{s}

_{bo} = _{s}

_{bo}, and the point _{bo} +

Work line plotting in the

The work line equation has the form

_{bo})(_{be} – _{bo})/(_{be} – _{bo}) + _{bo}. (5)

_{bo} < _{s}_{bo}+ Δ_{bo}. After simple transformations (4) and (5), assuming that Δα= α – α_{bo}, the value a is calculated by the formula

α = (α_{bo} + Δ

The value α_{bo} is determined in accordance with the formula

α_{bo} = 1.17^{0.35} / ^{0.15}(ρ^{0.15}, (7)

where ρ

_{0} < _{bo} < _{s}

α = α_{bo} (1 – _{bo})^{1.35}, (8)

_{bo} = –0.573^{0.7}(^{0.3}, (9)

following simple transformations, we get

_{bo}(1 – α^{0.74}/α_{bo}). (10)

_{pred} during surface boiling (_{bo} < _{s}

_{pred} =

The presented model development procedure suggests monotonous variation of parameters.

The changes in the work line position for the point

an abrupt reduction of the primary circuit pressure;

an abrupt temperature increase (Δ T);

an abrupt pressure (Δ P) and power (Δ Q) increase.

The physical meaning of the work line in the

In accordance with the work line _{K}_{bo} + Δ_{max} /2 (Δ_{max} = _{K}_{cr}, we shall identify two classes of the reactor core states in the event surface boiling starts:

there is no critical heat flux;

there is a critical heat flux.

We shall take entropy as the system state parameter.

Assuming that the distribution of the indicator _{bo} for class 1 and _{K}^{2}, Fig. _{cr}, where the distribution density functions intersect, corresponds to the entropy of the system in a condition for which the probabilities of the critical heat flux taking place and being absent are equal, that is, defines _{cr}.

The coordinates of the point _{cr}, _{cr}) on the work line (see Fig.

_{cr} = (_{bo} + _{K}_{bo} +

Δ_{bo}, Δ_{cr} – _{bo}, make it possible to calculate a_{cr}, _{cr}, and _{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.

Thermodynamic process of heat supply for a heat engine or for a nuclear reactor in design conditions (a) and for a nuclear reactor in emergencies (b)

Plotting of the critical point

Functions of the entropy distribution density for two classes of situations: 1 – with no critical heat flux; 2 – with a critical heat flux

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 second-order differences of the entropy with Δ

Δ^{2}^{2}/Δ^{2}

where Δ^{2}

We shall assume that the _{Ts}

_{Ts}_{bo} + _{Ts}_{bo}) =

_{bo} = _{Ts}_{bo}). (15)

In the event _{Ts}

A neural network with one perceptron is proposed to be used for the rapid identification of the channel state. The value _{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 _{cr} will correspond to the zero class, and those after _{cr} will correspond to class 1, it is possible to classify the given vector of the values (

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.