Corresponding author: Yury А. Kazansky (kazansky@iate.obninsk.ru)

Academic editor: Yury Korovin

It is assumed by the authors of the present paper that with growing contribution of nuclear power in the production of electricity, nuclear power plants will be used to a higher degree in a manoeuvrable mode of operation rather than in the base-load mode. In other words, change of power from the nominal level to that of coverage of auxiliary loads will be becoming quite common and not so rare event as scheduled reactor shutdowns for fuel reloading or preventive works. There exist well-known problems in the use of nuclear reactors in the manoeuvrable operation mode, which include the task shared by all types of nuclear reactors. It is advisable to have a unified indicator weakly power-dependent and fairly easy to measure, which would make it possible to formulate the judgement about the nature of the transient processes within the entire power range and to assess the reactivity required for changing the power level by the preset value. Power reactivity coefficient (PRC) can be used as such indicator. Analysis was made of existing definitions and understanding of PRC in relevant references. It turned out that there is no generally accepted definition of the PRC. Based on the performed study, the following definition was suggested: the PRC is the ratio of the low reactivity introduced into the reactor to the power increment at the end of the transient process. It is assumed here that variation of reactivity is dependent on the energy released in nuclear fission but is not related to the changes of reactivity induced by feedback signals in the automatic reactor power control system.

Analysis of the relationship between the PRC and temperature coefficients and technological parameters associated with the steady-state control program was performed taking the above suggested definition into account. PRC calculations were performed using the simplest model of VVER-1000 type power reactor. It was found that PRC is weakly power-dependent.

The purpose of the present study is to investigate dependence of PRC on the temperature reactivity effects and on the technological parameters associated with the steady-state control program of the power unit, using the example of VVER-1000. Effects of PRC on the static and dynamic power reactor operation modes are analyzed.

Statics and dynamics of nuclear power reactors are mainly determined by their intrinsic feedback links reflecting the effects on the reactivity of temperature and pressure in the reactor as well as the nuclear physics properties of materials in the reactor core dependent on them.

It is accepted to characterize the extent of the effects produced by separate process parameter (PP) by the respective coefficient of reactivity (CR), most often the temperature reactivity coefficient (TCR) and pressure coefficient of reactivity (PrCR) of, for instance, reactor coolant, fuel or moderator.

Mathematically CRs are expressed in the form of partial derivative of reactivity with respect to the PP variation of which produces effect on reactivity. Physically, for instance, both TCR and PrCR mean the ratio of the variation of reactivity to the small variation of temperature or pressure causing the reactivity variation with all other factor influencing reactivity remaining constant.

Equations of reactor dynamics containing equations for intrinsic feedbacks written taking CR into consideration provide the most exhaustive description of the dynamic and static modes of power reactor operation, which is demonstrated in the fundamental studies of NPP reactors in Russia (

Use of such operational mode of nuclear reactors is complicated by a number of issues associated with process parameters and with neutron physics characteristics. One of such issues is the necessity to look for surplus durability margin of structural materials, stability of which against neutron flux and temperature decreases in the conditions of frequent variation of power level as compared to constant neutron flux density (Ovchinnikov and et al. 2012). This is especially important for fast reactors the most important performance indicator of which, namely, the fuel burnup depth, is limited by the stability of original physical and technical characteristics of fuel and structural materials under irradiation by limiting values of neutron flux (

At the same time, practical application of TCR and PrCR, for instance, for prompt estimation of the reactivity worth required for preforming the planned maneuver of the reactor power is associated with certain difficulties often caused by the lack of timely updated information about the required process parameters. For example, reactor fuel temperature is not directly controlled by standard measuring channels which makes difficult application of fuel TCR in the calculations of thermal effects of reactivity of the reactor using fuel temperature.

PCR characterizing power effects of reactivity (PER) as the combined result of all effects of reactivity (

Let us examine to what extent PCR can serve as such characteristic taking into consideration the features of light water reactors (VVER-1000).

In fact, knowing the reactor PCR a_{w}_{1} to _{2} can be represented in the following form:

Besides the above, a_{w}

If it happens that a_{w}_{w}_{2} – _{1}). If a simple analytical expression for a_{w}

The main purpose of the present study is the investigation of different PCR aspects and representation of simplified analytical model for forecasting reactor reactivity behavior in static operational modes.

Often the description of reactor characteristics is limited by the generation of a table containing, among other coefficients of reactivity, the value of PCR or the power coefficient of reactivity as the self-evident concept. At the same time different and sometimes contradictory interpretations and definitions of PCR can be found in the reference sources (which, as a rule, are recommended as teaching aids). The most straightforward and widely spread is the PCR definition (

PCR was defined in (

Several methods for measurement of PCR on the operating power unit follow from the above definition: for instance, the method with automatic power control system switched off. Small reactivity is introduced in the reactor with APC switched off. Transient is initiated (power, fuel and coolant temperature, etc. are changing), after which a new power level is stabilized (PCR is assumed to be negative, the reactor is assumed to be stable). Ratio of the introduced reactivity to the variation of power is accepted as the estimated PCR value at the given power level.

Another method with APC switched on in the operational mode with maintained neutron flux power (“N” mode) consists of variation of the preset power. In such case APC will automatically change the reactor power by introducing the required reactivity. Similar to the previous case, the ratio of introduced reactivity to the variation of power is accepted as the PCR value. The value of introduced reactivity is estimated in both cases according to the known calibration characteristics of reactor controls. The method for PCR measurement by small variation of the preset electric power of the automatic control system during operation of the APC in “T” operational mode can also be suggested. As in the previous cases PCR is calculated in the form of the ratio of variation of reactivity to the value of variation of power upon completion of the transient.

Based on the suggested definition of PCR (variation of reactivity caused by the integral effect on it by the variations of fuel temperature, coolant temperature and pressure (

where _{0} is the reactor power level; a_{pi}_{i}

Thus, in accordance with (1) PCR is the ratio of the variation of reactivity to the small deviation of power from the initial level in the established steady-state mode causing the PCR variation.

In accordance with the above definition PCR possesses a number of attractive features:

PCR is defined as the total derivative with respect to power which allows automatically taking into account its dependence on all PPs dependent on power and influencing reactivity in real conditions of operation of the power unit rather than in the conditions artificially created for stabilizing a number of PP;

It is not difficult to experimentally measure PCR in the conditions of reactor operation because implementation of special measures for maintaining other PPs of the power unit is not required;

PCR can be expressed through other CRs.

Thus, in normal operational modes on power levels from zero to nominal power coolant pressure is maintained constant and, consequently, the value of pressure effect of reactivity is insignificant compared to temperature effects of reactivity, and further discussion will be limited by us with two PPs influencing reactivity, i.e. with temperatures of fuel and coolant.

In this case Formula (1) will be as follows:

where a_{f}_{c}_{f}_{c}

Let us find total derivatives of fuel and coolant temperatures with respect to power for Formula (2). Simple model of heat exchange in the reactor core (Fig.

Model of heat exchange in the reactor core: _{ci}_{co}_{c}

Relations between process parameters can be represented within the framework of the suggested model in the following form:

_{f}_{f}_{c}

_{c} c_{pc}_{co}_{ci}

where _{f}_{f}_{c}_{f}_{pc}

If half-sum of coolant temperatures at the reactor inlet and outlet is used as the mean coolant temperature, i.e. _{c}_{co}_{ci}_{c}_{f}_{pc}

_{c}_{c} c_{pc}_{ci}_{c}_{c}_{ci}

_{f}_{f}^{–1} + (2 _{c} c_{pc}^{–1})] = _{f}_{f}_{c}_{c}_{ci}

From (5) we obtain:

And from (6) we correspondingly get:

Subsequent estimations of total derivative of fuel and coolant temperatures with respect to power are possible only with SCP of the power unit taken into account.

SCPs with constant flow rate _{c}_{c}

As it follows from (

_{ci}_{c}_{0} + _{ct}w

where _{c}_{0} is the initial coolant temperature, K; _{ct}

In particular, it follows from (5) for SSCP with constant average coolant temperature that _{ct}_{c}c_{pc}^{–1} < 0. Coefficient _{ct}_{ci}_{ct}_{nom}) (

Having differentiated (5), (6) and (9) in accordance with (7) and (8) and substituting the obtained derivatives in (2) we obtain PCR in the following form:

It follows from the obtained expression that PCR depends on SCP. In particular, for SCP with _{c}

a_{w}_{f}_{f}_{c}

If _{ci}

It follows from expressions (10) and (11) that PCR depends on the current level of reactor power if remaining coefficients of reactivity have the same type of dependence.

Besides the above, coolant flow rate produces effect on a_{w}_{c}_{c}

Calculation code was written in SciLab environment for estimation of PCR dependences for widely spread SCPs during operation with four, three and two cooling loops of the primary cooling circuit representing (10) for the example of VVER-1000 under typical assumptions for reactor core models with lumped parameters:

Half-sum of coolant temperatures at the reactor inlet _{ci}_{co}

There is no non-uniformity of coolant flow rate and energy output in the reactor core;

Parabolic distribution of fuel temperature in the fuel pin is valid, i.e. mean fuel temperature exceeds the external temperature of the fuel rod by the value equal to two thirds of the maximum temperature differential inside the fuel rod.

Coolant heating in the reactor core is calculated as follows:

D_{c}_{pc}_{c}_{c}_{c}

Maximum fuel temperature inside the fuel pin [Kirillov P.L., 2010] is equal to:

_{f}_{max} = _{c}_{v}r_{f}_{eff}) + _{v}r_{f}^{2}/(4l

where _{v}^{3}; _{f}_{eff} is the effective heat transfer coefficient, W/(K∙m^{2}); l_{f}_{eff} is calculated according to Formula (

where a is the heat transfer coefficient, W/(K∙m^{2}); _{f}_{w}_{g}_{c}_{g}

Simplified formulas from (_{c}_{h}_{c}^{0,8}Pr_{c}^{0,4}; Reynolds number Re_{c}_{h}_{c}_{c}_{c}c_{pc}^{–2})^{0,15}; _{h}_{f}^{2} – 1) is the hydraulic diameter, m; _{f}_{c}_{c}^{2}/s are the coefficients of dynamic and kinematic viscosity of the coolant, respectively.

Linear approximation within the interval of operation al temperatures with limiting values equal to –1,0∙10^{–4} and –1,5∙10^{–4} K^{–1} (_{c}_{c}^{–4} and –2,0∙10^{–4} K^{–1} (

Results of calculation of PCR depending on the reactor power with fixed coolant flow rate are presented in Figs

Calculated dependences of PCR on reactor power for the given flow rates for SCP with constant coolant temperature at the reactor core inlet: a) for nominal coolant flow rate _{c}_{nom}; b) for operation with three cooling loops (_{c}_{nom}); c) for operation with two cooling loops (_{c}

Calculated dependences of PCR on reactor power for the given flow rates for SCP with constant steam throttle pressure (_{2} = const): a) for nominal coolant flow rate _{c}_{nom}; b) for operation with three cooling loops (_{c}_{nom}); c) for operation with two cooling loops (_{c}_{nom}) of the primary cooling circuit.

Calculated dependences of PCR on reactor power for the given flow rates for SCP with constant mean coolant temperature in the reactor core: a) for nominal coolant flow rate _{c}_{nom}; b) for operation with three cooling loops (_{c}_{nom}); c) for operation with two cooling loops (_{c}_{nom}) of the primary cooling circuit.

Analysis of the obtained calculated dependences demonstrates (see Figs

Averaged PCR values within the power range of 10 – 100% _{nom}, as well as maximum deviations of PCR from average value obtained on the basis of Figs

PCR values averaged over the power and maximum deviations from average value.

SSCP | Coolant flow rate, % _{c}_{nom} |
Average value 1/(%_{nom}) |
Maximum deviation Da_{w av}, % |
---|---|---|---|

_{ci} |
100 | –1.13∙10^{–4} |
3.8 |

75 | –1.24∙10^{–4} |
2.0 | |

50 | –1.43∙10^{–4} |
0.3 | |

_{2} = const |
100 | –1.25∙10^{–4} |
1.2 |

75 | –1.34∙10^{–4} |
1.9 | |

50 | –1.52∙10^{–4} |
2.7 | |

_{c} |
100 | –8.95∙10^{–5} |
6.6 |

75 | –9.18∙10^{–5} |
5.2 | |

50 | –9.54∙10^{–5} |
3.6 |

It follows from the data in the Table that for constant coolant flow rate in the primary cooling circuit dependence of PCR on power is fairly weak and does not exceed 10% within the whole range of its variation, which is comparable with accuracy of the performed calculations of heat exchange in the reactor core. Therefore, PCR can be regarded in the first approximation as constant and not dependent on the reactor power.

Reduction of coolant flow rate due, for instance, to the operation of the OLD system, results in the increase of PCR absolute value which, in turn, increases self-regulation properties of the reactor and produces favorable effect on the power unit safety.

More noticeable variation of PCR (about 40%) takes place when SCP is changed, for instance, in the case of transition from SCP with constant steam throttle pressure to SCP with constant average coolant temperature in the reactor core. This fact must be taken into account in constructing combined SCPs, because change of settings of automatic control devices such as APC may be required.

Definition of PCR in accordance with (1) as the ratio of variation of reactivity to the small deviation of power from the preset level causing the variation of reactivity in the steady-state operation mode and its expression in the form of total derivative

As it has been already mentioned in the Introduction variation of reactivity in the transition of reactor facility from power level _{1} to power level _{2} is equal to:

Straightforward and easy to use in practical calculations expression is obtained for Dr = a_{w}

Dependence a_{w}

The required TCR can be calculated using measured data obtained during reactor start-up. As applicable to VVER-1000 in accordance with (_{T}_{f}_{c}_{c}_{T}_{f}

Since PCR in accordance with (1) is determined for steady-state mode of operation, then calculation of transients in the reactor on the basis of PCR does not offer special advantages as compared with initial dynamics equations. Nevertheless, the necessary conditions of reactor stability are incorporated in the dependence a_{w}_{w}_{w}

✩ Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2018, n.1, pp. 63–74.