Corresponding author: Sergey K. Podgorny (

Academic editor: Yury Korovin

Advanced pressurized water reactors are the main part of a new generation of nuclear power plant projects under development that provide cost-effective power production for various needs (

The authors of the article propose a method for calculating the temperature field in the core of a heterogeneous reactor (using the example of a pressurized water reactor), which makes it possible to quickly assess the level of temperature safety of various changes in the core and has the necessary speed for analyzing transients in real time.

This method is based on the energy equation for an equivalent homogeneous core in the form of a heat equation that takes into account the main features of the simulated heterogeneous structure. The procedure for recovering the temperature field of a heterogeneous reactor uses the analytical relation obtained in this work for the heat sink function, taking into account inter-fuel element heat leakage losses.

Calculations of temperature fields in the model of the PWR type reactor (

Attention to pressurized water reactors is explained not only by their prevalence as an energy source in various types of nuclear power plants (

All computing codes of heterogeneous cores are complicated, requiring significant computing power and imposing individual restrictions on the area of use. Another thing is a homogeneous core, which, moreover, is presented as a continuous medium. For such an core, the energy equation can be written in the form of the heat conduction equation (

In this work, the task is to build an algorithm for recovering the temperature field in a heterogeneous core in stationary and transient processes based on the results of calculating the temperature in an equivalent homogeneous core of a nuclear reactor.

For definiteness and concretization of the relations obtained below, let us consider the core of a PWR type reactor (

Geometrical parameters and composition of the fuel elements

FE component | Dimensions | Material | ||
---|---|---|---|---|

Diameter, mm | Thickness, mm | Length, mm | ||

Cladding | 9.14 | – | 3658 | Zircaloy 4 |

Gas gap | – | 0.157 | Helium | |

Fuel rod | 7.844 | – | Uranium dioxide |

The pressure of the coolant at the core inlet
_{in}^{in}_{2} and its mass flow rate
_{out}

The coolant exchange between neighboring fuel assemblies occurs on an equal-mass basis. The mass flow rate of the coolant in any fuel assembly does not change along its height

Let us define a virtual channel as an elementary FA channel, in the center of the square cross-section (with side
_{1}) of which there is a single fuel element with an outer cladding diameter
_{sh}

Volumetric heat releases
_{v}

In the calculations of changes in the temperature field

In the analysis of coolant temperature changes and determination of the sink term power, the core is considered as heterogeneous, composed of elementary channels, including discrete components “

A single elementary channel is a model of the corresponding fuel assembly in the analysis of the temperature field of the reactor core.

Heat and mass transfer between neighboring equivalent channels is described in the quasi-stationary approximation.

The core is described as a continuous medium with weighted average temperatures in the calculated cells. The heat conduction equation in the form (

where Φ = Σε_{i}_{ef} is the effective thermal conductivity of the simulated system W/(m∙°C); _{v}^{3}; _{v}._{st}^{3}; weighted average temperature in the calculated cell of the calculated volume _{i}^{*}_{i}, ε^{*}_{i} = ε_{i}/Φ; ε_{i} = (ρ∙_{i}, where ρ,

The analysis showed that for the core of a pressurized water reactor, it is possible to use the relation for determining the heat removal from the surface of a fuel element in a non-stationary process, obtained in (Kuzevanov and Podgorny 2019) for the core of a high-temperature gas-cooled reactor:

_{v.st}^{0}(^{~} + ^{~}}, (2)

where σ = ^{2}; ^{3}; ^{0} is the heat flux density per unit area of the fuel element, W/m^{2} (index “0” is the stationary (initial) state); ^{~} is the a function of time, spatial coordinates (^{~} is the function reflecting the influence of boundary conditions.

One of the features of using relation (2) is the need to determine the true heating of the coolant in each elementary channel, taking into account thermal leakage between them, which is possible only if the design and hydrodynamic features of the heterogeneous core are considered.

With the adopted physical model of the core taken into account, the heat balance equation for the channel “

where Δ_{j}_{2},_{j}_{j}_{n}_{,j}〉 is the average density of the heat flux from other fuel assemblies, W/m^{2}; _{1} is the cross-size of a cell containing one fuel element, m;

For an arbitrary cross section _{n,j}_{n}_{n,j.}_{1} – _{n,j.}_{2}. Assume that the components of the heat flux _{n,j.}_{1} and _{n,j.}_{2} through the virtual side surface of the channel can be represented as:

where λ_{ef}_{2} is the effective distance between adjacent fuel assemblies, m.

And now we shall determine the effective thermal conductivity coefficient λ_{ef} based on the following considerations. Let us assume that the change in the intensity of heat transfer on the heating surface during the transition from the laminar flow regime to the turbulent one is directly related to the general change in the heat-conducting properties of the medium. Then we obtain:

α_{l}_{t}∙λ/λ_{ef}, (5)

where α and λ are the are the coefficients of heat transfer and molecular thermal conductivity, respectively (the indexes “

For the laminar regime on the stabilized section in the round pipe, the solution of the integral Lyon relation for a laminar fluid flow leads to the equality Nu_{l}

Extending the relation Nu_{l}_{e}

λ_{ef} = α_{t}_{e}

The transformation of equation (3) into a system of algebraic equations for the connection of flow, hydraulic and thermodynamic parameters of channels, convenient for analysis, was carried out using B. Petukhov’s formula for calculating Nu_{t}

1 ≥

Here ξ are friction resistance coefficients;

_{j}_{m}_{–1}_{m}_{–1}/ξ_{m}G_{m}_{1} = (^{*})/^{*}_{2}/_{1} is the heat exchange constant between fuel assemblies; ^{*}_{2} = _{2}^{*} = _{t}^{2} is the number of fuel elements in a square fuel assembly;

The system of equations (7) is supplemented by a system of equations for the pressure drop in a group of identical fuel assemblies in the form of the Darcy-Weisbach equations (

For the system of interconnected channels, it is proposed to determine ^{0}_{v}_{.st} as follows:

^{0}_{v.st} = ^{0} – ^{0}_{2}), (9)

where for a square lattice of fuel elements arrangement:

Note that in relation (10) _{l}_{1} and _{l}_{2} are the linear thermal resistance of the cladding (including the gas gap) and heat transfer, respectively.

Using expression (9) in the equation

div(λ_{ef}^{0}_{v} – ^{0}_{2}) = 0 (11)

together with the system of equations (7) makes it easy to determine the stationary temperature distribution in the reactor core.

The dimensionless time function _{τ} is included in the relations for determining ^{~} and ^{~} (2) (Kuzevanov and Podgorny 2019, ^{0}(τ = 0), ^{∞}(τ → ∞ after exposure to disturbing factors), _{τ} = ^{*}_{τ} = (^{0})/(^{∞}– ^{0}), which is confirmed by calculations of the temperature fields of a gas-cooled reactor (Kuzevanov and Podgorny 2019,

Analytical and computational studies conducted by the authors have shown the possibility of using the following dependencies when calculating the function _{τ} for cores of PWR reactors:

The following notations are used here:

_{i}_{sh} is the temperature of the outer surface of the fuel element cladding.

The results of calculating the weighted average temperature when the core is represented as an equivalent homogeneous medium formed the basis for the procedure for recovering the temperature field in the elements of any calculation cell, i.e., the coolant, cladding and fuel. In the coolant, the temperature field was not detailed; only its average temperature in the cross section of the elementary channel and the equality of coolant and cladding temperatures on the outer surface of the fuel element were taken into account. It was assumed that the temperature profile in the fuel cladding remains logarithmic, while in the fuel it was described by a power function during the entire transient process. Within these model approximations, the procedure for recovering the temperature field in any calculated cell of the core looks quite simple.

Indeed, at the time τ after the start of the transient process in the core, caused by an abrupt change in any of the parameters or several parameters that affect the temperature distribution in the core, the following fields are directly known as a result of calculating the equivalent homogeneous core:

average temperatures
_{2};

values of the heat sink function
_{v.st}

weighted average temperature

Neglecting the thermal inertia of the thin cladding, we additionally calculate the temperature on the outer _{sh}_{sh}

_{sh} = _{2} + _{v.st}_{1}^{2}∙_{l}_{2}/π; _{sh}_{sh}_{v.st}_{1}^{2}∙_{l}_{1}/π (14)

and find the average value _{sh}_{2}, _{sh}

The maximum temperature value in the fuel of the calculation cell _{f}

_{f}^{max} = 2_{f}_{sh}. (15)

We considered a calculation version of the core model, which consists of _{1} (

In terms of thermal, structural, flow and temperature characteristics, the calculated core corresponds to the PWR core (^{2} = 1) of ^{2} = 289 identical elementary channels in a real fuel assembly.

Figs

Comparison of the average core temperature with thermal power surges by 50%, coolant mass flow rate by 20%, and coolant temperature at the core inlet by 20%: 1 – CFD simulation; 2 – developed algorithm

Comparison of the average temperature in the core with a thermal power surge by 50%: 1 – CFD simulation; 2 – developed algorithm

Figs

Comparison of the recovered temperature fields of the fuel elements with the results of CFD simulation for a time of 5 seconds from the beginning of the transient process shown in Fig.

Comparison of the recovered temperature fields of the fuel elements with the results of CFD simulation for a time of 10 seconds from the beginning of the transient process shown in Fig.

The proposed method for recovering the temperature field of a heterogeneous reactor does not claim to increase the level of detail of the temperature distribution in the core components in comparison with the resulting description of the temperature field using CFD simulation. However, in some cases, the authors’ approach described in this paper can be useful, since it has the following advantages:

mobile availability of the computing power required for the calculations;

short time for the complete calculation of the temperature distribution in the local region of the core of interest to the researcher, i.e., approximately two orders of magnitude less than when the basic CFD simulation algorithms are used; and

operational preliminary calculation of a set of options for structurally different cores to select a limited number of them for the purpose of subsequent refining analysis.

Note that the time of calculating the temperature field by the recovery method is less than the time of the transient process. In this case, such a computational procedure can be an element of a complex program that describes the dynamics of the reactor circuit, e.g., in the software package of a nuclear power plant simulator. In addition, it is possible to use the recovery algorithm in the control systems of

^{th}International Scientific Conference “The Latest Research in Modern Science: Experience, Traditions and Innovations”. Morrisville, North Carolina, USA, 7–8 July, 20–31.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2022, n. 1, pp. 54–65.