Corresponding author: Sergey K. Podgorny (serkonpod@gmail.com)

Academic editor: Boris Balakin

Positive effect of profiling the gas-cooled reactor core within the framework of the GT-MHR project was investigated in (

The present paper is dedicated to the investigation of development of transients in gas-cooled nuclear reactor core subject to the implementation of different principles of core profiling.

Investigation of transients in reactor core represents complex problem, solution of which by conducting direct measurements is beyond the resources available to the authors. Besides the above, numerical simulation based on advanced CFD software complexes (

The algorithm for calculating temperature fields using the model where the reactor core is represented as the solid medium with gas voids was developed by the authors and the assumption was made that heat transfer due to molecular heat conductivity can be described by thermal conductivity equation written for continuous medium with thermal physics parameters equivalent to respective parameters of porous object in order to get the possibility of obtaining prompt solutions of this type of problems.

Computer code for calculating temperature field in gas-cooled reactor in transient operation modes was developed based on the suggested algorithm. Proprietary computation code was verified by comparing the results of numerous calculations with results of CFD-modeling of respective transients in the object imitating the core of gas-cooled nuclear reactor. The advantage of the developed computer code is the possibility of real-time calculation of evolution of conditions in complex configurations of gas-cooled reactor cores with different channel diameters. This allows using the computer code in the calculations of transients in loops of reactor facility as a whole, in particular for developing reactor simulators.

Results are provided of calculations of transients for reactor core imitating the core of gas-cooled nuclear reactor within the framework of GT-MHR project performed for different approaches to profiling coolant mass flow. Results of calculations unambiguously indicate the significant difference of temperature regimes during transients in the reactor core with and without profiling and undeniable enhancement of reliability of nuclear reactor (Design of the Reactor Core 2005, International Safeguards 2014, Safety of Nuclear Power Plants 2014) with profiling of coolant mass flow in the reactor core as a whole.

Distributions of flowrate and thermal parameters of coolant inside cooling channels of high-temperature helium-cooled nuclear reactor were investigated in details in (

Coolant mass flowrates in parallel channels;

Coolant heating in the channels;

Maximum wall temperatures of cooling channels.

Reactor operation on nominal power in steady-state operation mode was addressed.

In the event of perturbation with variation of either neutronics parameters (neutron field) or coolant parameters (flowrate, coolant temperature at the reactor core inlet) transient process develops with establishment of new temperature field in the reactor core. The present study is dedicated to the investigation of effects of coolant mass flowrate profiling on the variation of temperature in the reactor core during transients.

Physical and mathematical models of transients in gas-cooled nuclear reactor are formulated, calculation results are presented and their comparison with results of CFD-modeling is made using the example of GT-MHR nuclear reactor core.

Nuclear реактор within the framework of GT-MHR reactor design project (_{0} = 320 kg/s in nominal power operation mode, coolant temperature at the core inlet _{in} = 491^{о}C.

In the model representation the core is examined as the solid medium (index 1) with gas voids (index 2). Solid part is multi-component consisting of fuel, graphite and metal; single-component gaseous part consists of helium.

It is assumed that heat transfer due to molecular thermal conductivity can be described as for the continuous medium with thermal physics parameters equivalent to respective parameters of the porous object.

The following is accounted for in the model:

Thermal conductivity with varying density ρ and varying thermal conductivity coefficient λ of reactor core components;

Heat generated by nuclear reactions is represented as distributed internal heat sources q v ;

Heat dissipated by coolant from the cooled down surface is represented as internal heat sinks q v.t .

The following assumptions are made:

Averaged temperature Т within the calculation section (cell) is determined by the conditions of equivalent continuous medium;

Temperatures of solid and gaseous components differ from each other but, however, are linked with the calculated average temperature of the equivalent continuous medium;

Variation of averaged parameters of the continuous medium in tangential direction can be neglected; in this case reactor core is represented in two-dimensional approximation;

Average porosity (ε) of the equivalent continuous medium within the reactor core volume is equal to the ratio of the volume of gaseous component V2 to the total volume V of the core;

Local porosity ε is the function of coordinate;

Variation of porosity due to temperature effects can be neglected;

Conditions of gaseous component are described by equation written for ideal gases;

Pressure differentials P in the system are small – thermodynamic process during gas flow can be regarded as isobaric in the calculation of thermal physics parameters of the gas.

Basic elements in the model of heat transfer are following:

Thermal conductivity equation in the system with variable thermal physics parameters is the main energy balance equation;

Displacement (flow) of the gaseous component is taken into account as the transfer of heat represented in the form of source (sink) member in the thermal conductivity equation;

Due to the significant thermal inertia of the solid component variation of parameters of the gaseous component is calculated in quasi-stationary approximation.

Thus, the aggregated heat and mass transfer in the core of the nuclear reactor is represented in the model as the superposition of the non-stationary process of thermal conductivity and the quasi-stationary process of transfer (removal) of heat by the variable gas flow.

Let us write the thermal conductivity equation for the reactor core represented as the continuous medium with internal heat sources and sinks and with variable thermal physics properties in the following form (

where _{1}_{1}(1 – ε) + ρ_{2}_{2} ε; _{1}, _{2} are the enthalpies of the solid and gaseous components, respectively, J/K; ρ_{1} is the density of the multi-component solid fraction, kg/m^{3}; ρ_{2} is the helium density, kg/m^{3}; τ is the time, s; _{v}_{v.t}^{3}; λ_{ef}_{ef}

After transformation of the left side of Equation (1) we obtain

where the component _{p.}_{2}ρ_{2} + _{2}∙δρ_{2}/δ_{2} = 0 in the case of zero value of _{2} at the point _{1} is the average specific thermal capacity of the multi-component solid fraction within the temperature range under study, J/(kg∙Κ); _{p.}_{2} is the isobaric specific thermal capacity of helium (_{p.}_{2} = const), J/(kg∙Κ).

Differential equation

Describes the heterogenous nuclear reactor core cooled with gas coolant as the equivalent core with distributed solid and gaseous components without internal boundaries, within the volume of which heat sources and sinks are positioned.

Quite naturally the function _{v.t}

It is evident that the following condition is valid in the calculation cell

(_{v.t}_{i}_{i}q_{i}_{.d}, (3)

where σ_{i}_{i}/V_{i}_{i}_{i}^{3} and the heat exchange surface corresponding to this volume, m^{2}, respectively; _{i}_{.d} is the heat flux density per unit surface, W/m^{2} (additional index

The task of determination of the heat sink function is reduced to the task of determination of effective thermal flux density on the walls of cooling channels in non-stationary process.

In steady-state mode _{d} = _{n}_{n}

where ^{2}; _{boun} is the total heat flux on the external boundary of the reactor core, W.

During the transient _{d} ≠ _{n}

Let us write for arbitrary cooling channel of the reactor core the equation binding the coolant temperature _{2}._{d} averaged over the flow cross-section with normal heat flux density on the channel wall _{d}:

where

Following the above made assumptions the equation below is satisfied

_{d} = α(_{st.d} – _{2.d}), (6)

where α is the average heat transfer coefficient along the channel length for the case when gaseous coolant flows inside round channel with diameter ^{2}∙K; _{st.d} is the effective value of the channel wall temperature averaged over the channel perimeter at the point with respective coordinate

Differential equation for the determination of _{d} follows from Equation (5) taking expression (6) into account:

_{1} = π_{p.}_{2} is the parameter constant along the length of the channel in question following the assumptions made.

The function having the following form is the solution of Equation (7):

where _{2} is the integration constant determined by the conditions at the coolant inlet in the reactor core.

For determining _{d} using the relation (8) reflecting the connection with channel wall temperature _{st.d} it is necessary to know the functional dependence of this temperature on the coordinate and time.

The assumption of validity of the following relation was made within the framework of the suggested model

(_{st.} – _{st.0}) / (_{st.1} – _{st.0}) = _{τ}. (9)

Here index _{st.1} is the channel wall temperature averaged over the perimeter in newly established steady-state mode with modified parameters which affected the temperature regime; _{τ} is the function of time and radial coordinate not dependent on the coordinate

Assumption (9) is made based on the following considerations. Firstly, the reactor core structure does not change along any cooling channel (along the axial coordinate

Solution of Equation (8) taking into account the expression (9) allows finding the form of the function _{d} = _{d}(

Let us include _{2}^{in} in the list of perturbation factors influencing the temperature regime of the reactor core. Let us denominate the ratio of the new steady-state values of these parameters to the initial ones as follows:

_{q}_{1}/_{0}; _{α} = α_{1}/α_{0}; _{G}_{1}/_{0}; _{T}_{2.1}^{in}/_{2.0}^{in}. (10)

It follows from expression (9) that

_{st.d} = _{st.0} + (_{st.1} – _{st.0}) _{τ}. (11)

Let us take into consideration that stationary values _{st.0} and _{st.1} are determined as

Let us accept the “cosine” energy release dependence along the height of the core. If the coordinate origin is defined at the distance δ from the reactor core inlet we will have the following form of function _{0}(

_{0}(_{0}^{max} sin (π_{e}), (13)

where _{0}^{max} is the heat flux density on the wall channel in the middle of the channel along its height in the initial steady-state process; _{e} =

Let us note that with heat release density varying over the reactor core cross-section heat flux density _{0} in the steady-state process is the function of two coordinates:

_{0} = _{0}(_{0}^{max}(_{e}).

We integrate the first term of summation braced in figure brackets in Equation (8) and determine the integration constant _{2} from the condition of satisfaction of the following equation at the coolant inlet in the reactor core:

_{0}(δ,_{0}[_{st.0}(δ,_{2.0}^{in}

After not complicated transformations we obtain the following form of function _{d}(

where

In their turn,

Now it is easy to write down the expression for the determination of the heat sink function during the transient for calculation cell

where _{i}_{j}_{i}_{j} /V_{i}_{i}_{j}_{i}

According to its physical meaning the function _{τ} reflects the local thermal inertia of the reactor core.

We assume that function _{τ} cannot be expressed analytically in the general case and determine the function in arbitrary point as

_{τ} = (_{0}) / (_{1} – _{0}), (19)

where _{0} and _{1} are the respective solutions of the stationary equation

With boundary conditions identical to the initial (index

Numerical solution of Equation (2) gives the values of average temperatures of the equivalent continuous medium in the calculation cells. Coolant temperatures averaged over the volume of gaseous component in all cooling channels or in part thereof incorporated in the cells, as well as wall temperatures averaged over the perimeter of the channels are determined in each of the cells in the course of solution. Values of local maximum temperatures ^{max} in each of the cells are additionally determined for the purpose of obtaining the whole picture of the temperature regime of the reactor core both in steady-state and in transient processes. Calculation expressions for cell

_{i}_{j}^{max} = (_{st})_{i}_{j}_{i}_{j}^{max}, Δ_{i}_{j}^{max} = _{i}_{v}_{1}]}_{i}_{j}

where _{j}_{j}

_{j}_{e}/4)^{2}[2ln(_{e}/_{j}_{j}_{e})^{2} – 1]. (22)

Here, the value _{э} as the diameter of conventional isolated area with cooling channel _{j}

_{e} = [4^{1/2}, (23)

where ^{2};

The following numerical experiments were performed for the object imitating the core of gas-cooled nuclear reactor under the GT-MHR development project.

CFD-modeling of the reactor core requires significant computational capacity not available for the authors. Because of this reason comparison of results of calculation of transient processes using the methodology developed by the authors with those obtained using the algorithm with numerical CFD-modeling of the non-stationary process developing after exercising the perturbation was performed using the fragment of the reactor core (Fig.

Fragment of reactor core: 1 – channels with 15.88-mm diameter; 2 – channels with 12.7-mm diameter. Dimensions are given in mm; only a half of the fragment is shown in the figure because of mirror symmetry; total length of the fragment is equal to 1074.34 mm; height of the fragment is the same as the height of the reactor core equal to 7930 mm.

Qualitative and quantitative correspondence of calculated values of temperatures and those obtained as the result of CFD-modeling for the examined options of profiling the fragment of the reactor core and different types of perturbations are illustrated in Figures

Comparison of maximum and average temperatures in the reactor core with profiling under the condition of similar heating in case of power surge by 50%: 1 – values of maximum temperature obtained using the suggested algorithm; 2 – values of maximum temperature obtained in CFD-modeling; 3 – values of average temperature obtained using the suggested algorithm; 4 – values of average temperature obtained in CFD-modeling.

Comparison of maximum and average temperatures in the reactor core with profiling under the condition of similar wall temperature in case of drop of coolant mass flowrate by 50%: 1 – values of maximum temperature obtained using the suggested algorithm; 2 – values of maximum temperature obtained in CFD-modeling; 3 – values of average temperature obtained using the suggested algorithm; 4 – values of average temperature obtained in CFD-modeling.

Advantages of the core with profiling during the transient, essentially emergency, process as compared with the core without profiling is illustrated in Figures

Comparison of maximum temperature in the reactor core with different profiling conditions for the case of power surge by 50%: 1 – maximum temperature in the core without profiling; 2 – maximum temperature in the core with profiling under the condition of similar heating; 3 – maximum temperature in the core with profiling under the condition of similar maximum wall temperatures.

Similarly, in the case of sharp step-like drop of coolant mass flowrate by 50% with respect to the nominal flowrate value the difference in time for reaching maximum temperature equal to 1800 K amounts to about 150 s (see Fig.

It is evident that the core of nuclear reactor under the development GT-MHR project with coolant mass flowrate profiling is more reliable against the core without profiling in the situations requiring certain time for initiating response measures for accident localization.

Let us note (see Figs

Comparison of maximum temperature in the reactor core with different profiling conditions for the case of drop of mass flowrate by 50%: 1 – maximum temperature in the core without profiling; 2 – maximum temperature in the core with profiling under the condition of similar heating; 3 – maximum temperature in the core with profiling under the condition of similar maximum wall temperatures.

Algorithm was developed on the basis of which computer code was written for calculating temperature field in gas-cooled reactor during transient processes. Computer code was verified by comparing the results of numerous calculations with results of CFD-modeling of respective transient processes. The code has the advantage associated with the possibility of real-time calculation of evolution of conditions of gas-cooled cores with complex configuration containing channels with different diameters, which allows using the code in the calculations of transient processes in the cooling loops of nuclear facilities as a whole, in particular, in the development of reactor simulators.

Results of calculations of transient processes are presented for the reactor core imitating the core of high-temperature gas-cooled nuclear reactor under the GT-MHR development project for different conditions of profiling mass flowrate. Results of calculations unambiguously demonstrate significant differences between temperature regimes in the reactor cores with and without profiling during transient processes and indisputable enhancement of reliability of nuclear reactors with profiling coolant mass flowrate in the reactor core as a whole.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2019, n. 3, pp. 53–65.