**4. Method to determine the most strained cladding axial segment**

The amplitude of LHR jumps in AS occurring when the NR thermal power capacity *N* increases from 80% to 100% level, was estimated by the instrumentality of the RS code, which is a verified tool of the WWER-1000 calculation modelling (Philimonov and Мамichev, 1998). Using the RS code, the WWER-1000 core neutron-physical calculation numerical algorithms are based on consideration of simultaneous two-group diffusion equations, which are solved for a three-dimensional object (the reactor core) composed of a limited number of meshes.

Theory of Fuel Life Control Methods at Nuclear

The number of fuel assemblies

*H*3=84%.

FA group

maximum LHR max

АS

stays in the 31-st FA ( max

FA stays in the 69-th FA ( max

year (the algorithm 55–31–69–82).

the FA stays in the central 82-d FA ( max

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 213

The lowest control rod axial coordinates for *N*1=100% and *N*3=80% were designated *H*1=90% and *H*3=84%, respectively. That is when *N* changes from *N*1=100% to *N*3=80%, the lowest control rod axial coordinate measured from the core bottom changes from *H*1=90% to

It has been found using the RS code that the WWER-1000 fuel assemblies can be classified into three groups by the FA power growth amplitude occurring when the NR capacity

2 37 26 20, 42, 43, 46, 51, 53…57, 66…71, 80…84, 93…98,

When the eighth, ninth and tenth regulating groups are simultaneously used, the central FA (No. 82) as well as fresh fuel assemblies are regulated by control rods. But when using the A-algorithm, the tenth regulating group is used only. In this case, such a four-year FA transposition algorithm can be considered as an example: a FA stays in the 55-th FA (FE

The average LHR for *i*-segment and *j*-FA is denoted as *l i* , , *<sup>j</sup>* < > *q* . For all segments (*n* = 8) of the 55-th, 31-st, 69-th and 82-nd fuel assemblies, the values of *l i* , , *<sup>j</sup>* < > *q* have been calculated at power levels of *N*3=80% and *N*1=100% using the RS code. The

8 1.341 1.517 1.328 1.340 7 1.308 1.426 1.297 1.309 6 1.250 1.241 1.263 1.268 5 1.229 1.213 1.238 1.250 4 1.224 1.217 1.232 1.242 3 1.241 1.229 1.243 1.259 2 1.255 1.251 1.271 1.270 1 1.278 1.275 1.288 1.302 Table 6. The , , , , (100%)/ (80%) *li j li j* <> <> *q q* ratio values for fuel assemblies 55, 31, 69, 82.

FA number 55 31 69 82

*<sup>l</sup> q* = 236.8 W/cm, FA group 2) position for the first year – then the FA

*<sup>l</sup> q* = 250.3 W/cm, group 1) position for the second year – further the

*<sup>l</sup> q* = 171.9 W/cm, group 2) position for the third year – at last,

*<sup>l</sup> q* = 119.6 W/cm, group 2) position for the fourth

FA numbers (according to the core cartogram )

107…111, 113, 118, 121, 122, 144

increases from 80% tо 100% level – see Table 5 (Pelykh et al., 2010).

FA power growth, %

Table 5. Three groups of the WWER-1000 fuel assemblies.

, , , , (100%)/ (80%) *li j li j* <> <> *q q* ratio values are listed in Table 6.

1 6 28 31, 52, 58, 106, 112, 133

3 120 ≤ 25 all other fuel assemblies

The amplitude of LHR jumps was calculated for the following daily power maneuvering method: lowering of *N* from *N*1=100% to *N*2=90% by injection of boric acid solution within 0.5 h – further lowering of *N* to *N*3=80% due to reactor poisoning within 2.5 h – operation at *N*3=80% within 4 h – rising of *N* tо the nominal capacity level *N*1=100% within 2 h (Maksimov et al., 2009). According to this maneuvering method, the inlet coolant temperature is kept constant while the NR capacity changes in the range *N*=100–80%, and the initial steam pressure of the secondary coolant circuit changes within the standard range of 58–60 bar. It was supposed that the only group of regulating units being used at NR power maneuvering was the tenth one, while the control rods of all the other groups of regulating units were completely removed from the active core. The next assumption was that the Advanced power control algorithm (A-algorithm) was used. The WWER-1000 core contains ten groups of regulating units in case of the A-algorithm – see Fig. 8.

Fig. 8. Disposition of the WWER-1000 regulating units in case of the A-algorithm: (upper figure) the FA number; (middle figure) the lowest control rod axial coordinate (at 100% NR power level) measured from the core bottom, %; (lower figure) the regulating unit group number.

The amplitude of LHR jumps was calculated for the following daily power maneuvering method: lowering of *N* from *N*1=100% to *N*2=90% by injection of boric acid solution within 0.5 h – further lowering of *N* to *N*3=80% due to reactor poisoning within 2.5 h – operation at *N*3=80% within 4 h – rising of *N* tо the nominal capacity level *N*1=100% within 2 h (Maksimov et al., 2009). According to this maneuvering method, the inlet coolant temperature is kept constant while the NR capacity changes in the range *N*=100–80%, and the initial steam pressure of the secondary coolant circuit changes within the standard range of 58–60 bar. It was supposed that the only group of regulating units being used at NR power maneuvering was the tenth one, while the control rods of all the other groups of regulating units were completely removed from the active core. The next assumption was that the Advanced power control algorithm (A-algorithm) was used. The WWER-1000 core

contains ten groups of regulating units in case of the A-algorithm – see Fig. 8.

Fig. 8. Disposition of the WWER-1000 regulating units in case of the A-algorithm: (upper figure) the FA number; (middle figure) the lowest control rod axial coordinate (at 100% NR power level) measured from the core bottom, %; (lower figure) the regulating unit group

number.

The lowest control rod axial coordinates for *N*1=100% and *N*3=80% were designated *H*1=90% and *H*3=84%, respectively. That is when *N* changes from *N*1=100% to *N*3=80%, the lowest control rod axial coordinate measured from the core bottom changes from *H*1=90% to *H*3=84%.

It has been found using the RS code that the WWER-1000 fuel assemblies can be classified into three groups by the FA power growth amplitude occurring when the NR capacity increases from 80% tо 100% level – see Table 5 (Pelykh et al., 2010).


Table 5. Three groups of the WWER-1000 fuel assemblies.

When the eighth, ninth and tenth regulating groups are simultaneously used, the central FA (No. 82) as well as fresh fuel assemblies are regulated by control rods. But when using the A-algorithm, the tenth regulating group is used only. In this case, such a four-year FA transposition algorithm can be considered as an example: a FA stays in the 55-th FA (FE maximum LHR max *<sup>l</sup> q* = 236.8 W/cm, FA group 2) position for the first year – then the FA stays in the 31-st FA ( max *<sup>l</sup> q* = 250.3 W/cm, group 1) position for the second year – further the FA stays in the 69-th FA ( max *<sup>l</sup> q* = 171.9 W/cm, group 2) position for the third year – at last, the FA stays in the central 82-d FA ( max *<sup>l</sup> q* = 119.6 W/cm, group 2) position for the fourth year (the algorithm 55–31–69–82).

The average LHR for *i*-segment and *j*-FA is denoted as *l i* , , *<sup>j</sup>* < > *q* . For all segments (*n* = 8) of the 55-th, 31-st, 69-th and 82-nd fuel assemblies, the values of *l i* , , *<sup>j</sup>* < > *q* have been calculated at power levels of *N*3=80% and *N*1=100% using the RS code. The , , , , (100%)/ (80%) *li j li j* <> <> *q q* ratio values are listed in Table 6.


Table 6. The , , , , (100%)/ (80%) *li j li j* <> <> *q q* ratio values for fuel assemblies 55, 31, 69, 82.

Theory of Fuel Life Control Methods at Nuclear

time at daily cycle power maneuvering.

code (Suzuki, 2010).

estimated as

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 215

axial segment and above it the value drops in the sixth segment situated between the axial coordinates z = 2.19 and 2.63 m reflects the fact that the most considerable LHR jumps take place at the core upper region (see Table 6). Thus, taking account of the 55–31–69–82 fouryear FA transposition algorithm as well as considering the regulating unit disposition, on condition that the FE length is divided into eight equal-length axial segments, the sixth (counting from the core bottom) AS cladding durability limits the WWER-1000 operation

Growth of the water-side oxide layer of cladding can cause overshoot of permissible limits for the layer outer surface temperature prior to the cladding collapse moment. The corrosion models of EPRI (MATPRO-09, 1976) and MATPRO-A (SCDAP/RELAP5/MOD2, 1990) have been used for zircaloy cladding corrosion rate estimation. According to the EPRI model, the

where *dS dt* / is the oxide growth rate, µm/day; A = 6.3×109 µm3/day; *S* is the oxide layer thickness, µm; Q1=32289 cal/mol; R=1.987 cal/(mol K);*Tb* is the temperature at the oxide layer-metal phase boundary, K; COR is an adjusting factor which is added in the FEMAXI

According to the MATPRO-A model, the oxide layer thickness for a nucleate boiling flow is

where *S* is the oxide layer thickness, m; A = 1.5 (PWR); *t* is time, days; *Tb* is the temperature

The cladding failure parameter values listed in Table 7 have been obtained using the MATPRO-A corrosion model at COR = 1. If COR is the same in both the models, the MATPRO-model estimation of cladding corrosion rate is more conservative than the EPRImodel estimation, under the WWER-1000 conditions. Regardless of the model we use, the factor COR must be determined so that the calculated oxide layer thickness fits to experimental data. The oxide layer thickness calculation has been carried out for the described method of daily power maneuvering, assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm. The calculations assumed that, the Piling-Bedworth ratio was 1.56, the initial oxide layer thickness was 0.1 µm, the maximum oxide layer thickness was restricted by 100 µm, the radial portion of cladding corrosion volume expansion ratio was 80%. It has been found that the calculated cladding oxide layer thickness, for the WWER-1000 conditions and burnup *Bu* = 52.5 MW day / kg , conforms to the generalized experimental data obtained for PWR in-pile conditions (Bull, 2005), when

at the oxide layer-metal phase boundary, K; *S*0 is the initial oxide layer thickness, m.

A COR 1/3 3 3 <sup>0</sup> (4.976 10 exp( 15660 / ) ) (1 ) *S <sup>b</sup> t TS* <sup>−</sup> =× − + + , (17)

ω τ

A Q R COR <sup>2</sup> / ( / ) exp( / ) (1 ), <sup>1</sup> *<sup>b</sup> dS dt S* = −+ *T* (16)

ω τ( )

η= 1.

( ) exists at the fifth (central)

then it stayed in the 31-st FA position for the remaining fuel operation period, the

value reached 1.0 and the cladding collapse was predicted at τ < 2495 days with

The prediction shown in Table 7 that the largest value of

cladding corrosion rate for a bubble flow is estimated as

using the EPRI model at COR = - 0.431 – see Fig. 9.

Though the Nb-containing zirconium alloy E-110 (Zr + 1% Nb) has been used for many years in FE of WWER-1000, there is no public data on E-110 cladding corrosion and creep rates for all possible loading conditions of WWER-1000. In order to apply the cladding durability estimation method based on the corrosion and creep models developed for Zircaloy-4 to another cladding alloy used in WWER-1000, it is enough to prove that using these models under the WWER-1000 active core conditions ensures conservatism of the E-110 cladding durability estimation. Nevertheless, the main results of the present analysis will not be changed by including models developed for another cladding alloy.

The modified cladding failure criterion at NR variable loading is given as (Pelykh and Maksimov, 2011):

$$\alpha(\boldsymbol{\pi}) = A(\boldsymbol{\pi}) \;/\ A\_0 = \mathbf{1}; A(\boldsymbol{\pi}) = \bigvee\_{\ell=0}^{\star} \sigma\_{\varepsilon}^{\max}(\boldsymbol{\pi}) \, j\_{\varepsilon}^{\max}(\boldsymbol{\pi}) \, d\boldsymbol{\pi}; \ A\_0 \text{ at } \ \sigma\_{\varepsilon}^{\max}(\boldsymbol{\pi}\_0) = \boldsymbol{\eta} \, \sigma\_0^{\max}(\boldsymbol{\pi}\_0). \tag{15}$$

where ω τ( ) is cladding material failure parameter; τ is time, s; *A*( ) τ is SDE, J/m3; *A*<sup>0</sup> is SDE at the moment 0 τ of cladding material failure beginning, when max max 0 00 ( ) ( ); σ τ ησ τ *<sup>e</sup>* = max( ) σ τ *e* and max( ) *<sup>e</sup> p* τ are equivalent stress (Pa) and rate of equivalent creep strain (s-1) for the cladding point of an AS having the maximum temperature, respectively; max 0 σ τ( ) is yield stress for the cladding point of an AS having the maximum temperature, Pa; η is some factor, η≤ 1.

Assuming the 55–31–69–82 four-year FA transposition algorithm and η = 0.6 , the ω τ( ) values have been calculated by Eq. (15) using the following procedure: calculating max( ), σ τ *e* max( ) *<sup>e</sup> p* τ and max 0 σ τ( ) by the instrumentality of FEMAXI-V code (Suzuki, 2000); calculating *A*( ) τ ; determining the moment 0 τ according to the condition max max 0 00 ( ) ( ); σ τ ησ τ *<sup>e</sup>* = determining 0 0 *A A*≡ ( ) τ ; calculating ω τ( ) – see Table 7 (Pelykh and Maksimov, 2011).


Table 7. Cladding failure parameters ω τ( ) for the axial segments 4–7.

For the other axial segments No. 1–3 and 8, on condition that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, the ω τ( ) value was less than 1.0, i.e. there was no cladding collapse up to τ = 2495 days. For τ > 2495 days calculations were not carried out. For all the axial segments, on condition that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, it has been found that there was no cladding collapse up to τ = 2495 days with ω τ( ) = 1. At the same time, for all the axial segments, on condition that a FA stayed in the 55-th FA position for all fuel operation period, as well as on condition that a FA stayed in the 55-th FA position for the first year,

Though the Nb-containing zirconium alloy E-110 (Zr + 1% Nb) has been used for many years in FE of WWER-1000, there is no public data on E-110 cladding corrosion and creep rates for all possible loading conditions of WWER-1000. In order to apply the cladding durability estimation method based on the corrosion and creep models developed for Zircaloy-4 to another cladding alloy used in WWER-1000, it is enough to prove that using these models under the WWER-1000 active core conditions ensures conservatism of the E-110 cladding durability estimation. Nevertheless, the main results of the present analysis

The modified cladding failure criterion at NR variable loading is given as (Pelykh and

0 0 0 00

= == <sup>=</sup> (15)

at max max max max

are equivalent stress (Pa) and rate of equivalent creep strain (s-1) for the cladding

τ

( ) for the axial segments 4–7.

; calculating

 σ  τ ησ

τ

 τ ησ

> τ

is some factor,

0 σ

σ

η

( ) by the instrumentality of FEMAXI-V code (Suzuki, 2000);

ω τ

> ω τ

( ) = 1. At the same time, for all the axial

АS 4 5 6 7

 τ

0 00 ( ) ( );

η

is SDE, J/m3; *A*<sup>0</sup> is SDE at

 τ*<sup>e</sup>* = max( )

> η≤ 1.

( ) – see Table 7 (Pelykh and

( ) value was less than 1.0,

( ) is yield stress for the

= 0.6 , the

σ

 τ*e*

> ω τ( )

will not be changed by including models developed for another cladding alloy.

( ) ( ) / 1; ( ) ( ) ( ) ; ( ) ( ), *A A A p dA e e <sup>e</sup>*

 τ τ

of cladding material failure beginning, when max max

values have been calculated by Eq. (15) using the following procedure: calculating max( ),

according to the condition max max

360 0.063 0.151 0.190 0.175 720 0.598 0.645 0.647 0.547 1080 0.733 0.783 0.790 0.707 1440 0.788 0.838 0.848 0.779

For the other axial segments No. 1–3 and 8, on condition that a FA was transposed in

i.e. there was no cladding collapse up to τ = 2495 days. For τ > 2495 days calculations were not carried out. For all the axial segments, on condition that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, it has been found that there was no

> ω τ

segments, on condition that a FA stayed in the 55-th FA position for all fuel operation period, as well as on condition that a FA stayed in the 55-th FA position for the first year,

τ

 τ

( ) is cladding material failure parameter; τ is time, s; *A*( )

Assuming the 55–31–69–82 four-year FA transposition algorithm and

; determining the moment 0

ω τ

concordance with the 55–31–69–82 four-year algorithm, the

point of an AS having the maximum temperature, respectively; max

0

cladding point of an AS having the maximum temperature, Pa;

 τ  σ

 τ τ

Maksimov, 2011):

 τ

τ

max( ) *<sup>e</sup> p* τ

0 00 ( ) ( );

τ, days

τ

 τ

Table 7. Cladding failure parameters

cladding collapse up to τ = 2495 days with

 and max 0 σ

*<sup>e</sup>* = determining 0 0 *A A*≡ ( )

ωτ

the moment 0

and max( ) *<sup>e</sup> p* τ

> τ*e*

> > τ ησ

calculating *A*( )

Maksimov, 2011).

where ω τ

σ

σ

then it stayed in the 31-st FA position for the remaining fuel operation period, the ω τ( ) value reached 1.0 and the cladding collapse was predicted at τ < 2495 days with η= 1.

The prediction shown in Table 7 that the largest value of ω τ( ) exists at the fifth (central) axial segment and above it the value drops in the sixth segment situated between the axial coordinates z = 2.19 and 2.63 m reflects the fact that the most considerable LHR jumps take place at the core upper region (see Table 6). Thus, taking account of the 55–31–69–82 fouryear FA transposition algorithm as well as considering the regulating unit disposition, on condition that the FE length is divided into eight equal-length axial segments, the sixth (counting from the core bottom) AS cladding durability limits the WWER-1000 operation time at daily cycle power maneuvering.

Growth of the water-side oxide layer of cladding can cause overshoot of permissible limits for the layer outer surface temperature prior to the cladding collapse moment. The corrosion models of EPRI (MATPRO-09, 1976) and MATPRO-A (SCDAP/RELAP5/MOD2, 1990) have been used for zircaloy cladding corrosion rate estimation. According to the EPRI model, the cladding corrosion rate for a bubble flow is estimated as

$$dS \mid dt = \left(\mathbf{A} \;/\; S^2\right) \exp(-\mathbf{Q}\_1 \;/\; \mathbf{R} \; T\_b) \left(1 + \text{COR}\right),\tag{16}$$

where *dS dt* / is the oxide growth rate, µm/day; A = 6.3×109 µm3/day; *S* is the oxide layer thickness, µm; Q1=32289 cal/mol; R=1.987 cal/(mol K);*Tb* is the temperature at the oxide layer-metal phase boundary, K; COR is an adjusting factor which is added in the FEMAXI code (Suzuki, 2010).

According to the MATPRO-A model, the oxide layer thickness for a nucleate boiling flow is estimated as

$$S = \left(4.976 \times 10^{-3} \,\mathrm{A} \,\mathrm{t} \,\exp(-15660 \,/\, T\_b) + S\_0\right)^{1/3} \left(1 + \mathrm{COR}\right),\tag{17}$$

where *S* is the oxide layer thickness, m; A = 1.5 (PWR); *t* is time, days; *Tb* is the temperature at the oxide layer-metal phase boundary, K; *S*0 is the initial oxide layer thickness, m.

The cladding failure parameter values listed in Table 7 have been obtained using the MATPRO-A corrosion model at COR = 1. If COR is the same in both the models, the MATPRO-model estimation of cladding corrosion rate is more conservative than the EPRImodel estimation, under the WWER-1000 conditions. Regardless of the model we use, the factor COR must be determined so that the calculated oxide layer thickness fits to experimental data. The oxide layer thickness calculation has been carried out for the described method of daily power maneuvering, assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm. The calculations assumed that, the Piling-Bedworth ratio was 1.56, the initial oxide layer thickness was 0.1 µm, the maximum oxide layer thickness was restricted by 100 µm, the radial portion of cladding corrosion volume expansion ratio was 80%. It has been found that the calculated cladding oxide layer thickness, for the WWER-1000 conditions and burnup *Bu* = 52.5 MW day / kg , conforms to the generalized experimental data obtained for PWR in-pile conditions (Bull, 2005), when using the EPRI model at COR = - 0.431 – see Fig. 9.

Theory of Fuel Life Control Methods at Nuclear

Let us introduce a dimensionless parameter *I*

thickness and the increase in *Tclad* ,*in* (see Table 9).

time, days.

corrosion.

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 217

a b Fig. 10. The SDE as a function of time for the sixth axial segment:(1, 2, 3, 4) at COR = 2, 1, 0, -0.431, respectively; (а) the EPRI model corrosion; (b) the MATPRO-A model corrosion.

> C day 6

been calculated for the sixth segment, using the EPRI corrosion model – see Table 9.

Table 9. Сladding temperatures subject to COR for the sixth segment, the EPRI model

This shows that the effect of cladding outer surface corrosion rate (with COR) on the cladding SDE increase rate (see Fig. 10) is induced by the thermal resistance of oxide

It should be noticed that the metal wall thickness decrease due to oxidation is considered in the calculation of the SDE, as effect of the cladding waterside corrosion on heat transfer and mechanical behavior of the cladding is taken into account in the FEMAXI code. Since

0 <sup>10</sup> , *T clad in I T dt* −

where *Tclad in*, is the cladding inner surface temperature for an axial segment, °C ; and *t* is

Having analysed the described method of daily power maneuvering, the maximum cladding oxide layer outer surface temperature max *Tоx out* , during the period of 2400 days, as well as *I*(2400 days) and the 2400 days period averaged cladding inner surface temperature < > *Tclad in*, have

COR max *Tоx out* , , °C *I*(2400 days) < > *Tclad in*, , °C

2 349.2 0.951 396.2 1 349.5 0.947 394.5 0 349.6 0.938 390.7 -0.431 349.6 0.916 381.8

,

= ⋅ ° ⋅ (18)

Fig. 9. Cladding oxide layer thickness *S* subject to height *h*: (■) calculated using the EPRI model at COR = - 0.431; In accordance with (Bull, 2005): (1) zircaloy-4; (2) improved zircaloy-4; (3) ZIRLO.

The EPRI model at COR = - 0.431 also gives the calculated cladding oxide layer thickness values which were in compliance with the generalized experimental data for zircaloy-4 (Kesterson and Yueh, 2006). For the segments 5–8, assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, the maximum oxide layer outer surface temperature max *Tоx out* , during the four-year fuel life-time has been calculated (EPRI, COR = - 0.431) − see Table 8. Also, for the segments 5–8, the calculated oxide layer thickness *S* and oxide layer outer surface temperature *Tоx out* , subject to time τ are listed in Table 8.

The maximum oxide layer outer surface temperature during the four-year fuel life-time does not exceed the permissible limit temperature lim *Tоx out* , =352 °C (Shmelev et al., 2004).


Table 8. The maximum oxide layer outer surface temperature.

The same result has been obtained for the EPRI model at COR = 0; 1; 2 as well as for the MATPRO-A model at COR = - 0.431; 0; 1; 2. Hence the oxide layer outer surface temperature should not be considered as the limiting factor prior to the cladding collapse moment determined in accordance with the criterion (15). Though influence of the outer oxide layer thickness on the inner cladding surface temperature must be studied.

Having calculated the SDE by the instrumentality of FEMAXI (Suzuki, 2010), assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, it has been found for the sixth axial segment that the number of calendar daily cycles prior to the beginning of the rapid creep stage was essentially different at СOR = - 0.431; 0; 1; and 2. As a result, the rapid creep stage is degenerated for both the corrosion models at СOR = - 0.431 (Fig. 10).

Fig. 9. Cladding oxide layer thickness *S* subject to height *h*: (■) calculated using the EPRI model at COR = - 0.431; In accordance with (Bull, 2005): (1) zircaloy-4; (2) improved

not exceed the permissible limit temperature lim *Tоx out* , =352 °C (Shmelev et al., 2004).

Table 8. The maximum oxide layer outer surface temperature.

5 345.1 11.3 (342.3) 40.6 (344.8) 58.1 (328.2) 69.8 (316.7) 6 349.6 16.1 (347.6) 49.8 (349.4) 69.3 (332.6) 82.5 (320.1) 7 351.2 18.1 (350.0) 52.7 (351.0) 74.1 (336.1) 88.5 (323.0) 8 348.0 14.2 (347.9) 38.3 (346.9) 58.0 (335.6) 71.2 (323.3)

The same result has been obtained for the EPRI model at COR = 0; 1; 2 as well as for the MATPRO-A model at COR = - 0.431; 0; 1; 2. Hence the oxide layer outer surface temperature should not be considered as the limiting factor prior to the cladding collapse moment determined in accordance with the criterion (15). Though influence of the outer

Having calculated the SDE by the instrumentality of FEMAXI (Suzuki, 2010), assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, it has been found for the sixth axial segment that the number of calendar daily cycles prior to the beginning of the rapid creep stage was essentially different at СOR = - 0.431; 0; 1; and 2. As a result, the rapid creep stage is degenerated for both the corrosion models at СOR = - 0.431

oxide layer thickness on the inner cladding surface temperature must be studied.

The EPRI model at COR = - 0.431 also gives the calculated cladding oxide layer thickness values which were in compliance with the generalized experimental data for zircaloy-4 (Kesterson and Yueh, 2006). For the segments 5–8, assuming that a FA was transposed in concordance with the 55–31–69–82 four-year algorithm, the maximum oxide layer outer surface temperature max *Tоx out* , during the four-year fuel life-time has been calculated (EPRI, COR = - 0.431) − see Table 8. Also, for the segments 5–8, the calculated oxide layer thickness *S* and oxide layer outer surface temperature *Tоx out* , subject to time τ are listed in Table 8. The maximum oxide layer outer surface temperature during the four-year fuel life-time does

S, µm ( *Tоx out* , , °C )

360 days 720 days 1080 days 1440 days

zircaloy-4; (3) ZIRLO.

<sup>i</sup>max *Tоx out* , , °C

(Fig. 10).

Fig. 10. The SDE as a function of time for the sixth axial segment:(1, 2, 3, 4) at COR = 2, 1, 0, -0.431, respectively; (а) the EPRI model corrosion; (b) the MATPRO-A model corrosion.

Let us introduce a dimensionless parameter *I*

$$I = \frac{10^{-6}}{^\circ \text{C} \cdot \text{day}} \Big|\_{0}^{T} T\_{\text{clad},in} \cdot dt\_{\text{\textdegree}} \tag{18}$$

where *Tclad in*, is the cladding inner surface temperature for an axial segment, °C ; and *t* is time, days.

Having analysed the described method of daily power maneuvering, the maximum cladding oxide layer outer surface temperature max *Tоx out* , during the period of 2400 days, as well as *I*(2400 days) and the 2400 days period averaged cladding inner surface temperature < > *Tclad in*, have been calculated for the sixth segment, using the EPRI corrosion model – see Table 9.


Table 9. Сladding temperatures subject to COR for the sixth segment, the EPRI model corrosion.

This shows that the effect of cladding outer surface corrosion rate (with COR) on the cladding SDE increase rate (see Fig. 10) is induced by the thermal resistance of oxide thickness and the increase in *Tclad* ,*in* (see Table 9).

It should be noticed that the metal wall thickness decrease due to oxidation is considered in the calculation of the SDE, as effect of the cladding waterside corrosion on heat transfer and mechanical behavior of the cladding is taken into account in the FEMAXI code. Since

Theory of Fuel Life Control Methods at Nuclear

surface corrosion rate) increases.

medium-load FE in FA 55 (maximum LHR max

h → exploitation at *N* = 100 % for Δτ h, corresponding to

<< 1 Hz and CF=idem, then there was no decrease of 0

ν

Table 10. Change of cladding failure time depending on

taken into consideration and cladding creep strain rate ( ) *<sup>e</sup> p*

Calculation of the cladding failure beginning moment τ0 depending on

corresponds to the experimental results (Kim et al., 2007) in principle.

(η=0.4, AS 6). At the same time, when *N*=100 % =const (CF=1), the calculated 0

2005).

respectively(

ν

ν

comparison with the case

significantly – see Table 10.

ν

cladding life.

<< 1 Hz).

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 219

It is necessary to notice that line 1 in Fig. 11 was calculated using separate consideration of steady-state operation and varying duty. When using equation (2), the fatigue component has an overwhelming size in comparison with the static one (Novikov et al.,

Use of the MATPRO-А corrosion model under the WWER-1000 core conditions ensures conservatism of the E-110 cladding durability estimation (see Fig. 11). Growth rate of *A*(τ) depends significantly on the FA transposition algorithm. The number of daily cycles prior to the beginning of rapid creep stage decreases significantly when COR (cladding outer

Setting the WWER-1000 regime and FA constructional parameters, a calculation study of Zircaloy-4 cladding fatigue factor at variable load frequency ν << 1 Hz, under variable loading, was carried out. The investigated WWER-1000 fuel cladding had an outer diameter and thickness of 9.1 mm and 0.69 mm, respectively. The microstructure of Zircaloy-4 was a stress-relieved state. Using the cladding corrosion model EPRI (Suzuki, 2000), AS 6 of a

analysed (COR = 1, inlet coolant temperature *Tin=*const=287 °C). The variable loading cycle 100–80–100 % was studied for Δτ =11; 5; 2 h (reactor capacity factor CF=0.9): *N* lowering from 100 to 80 % for 1 h → exploitation at *N* = 80 % for Δτ h → *N* rising to *N*nom=100 % for 1

Hence, the WWER-1000 FE cladding durability estimation based on the CET model

CF 0.9 1

 , cycle/day 1 2 4 – τ0, day 547.6 547.0 549.0 436.6

In the creep model used in the FEMAXI code (Suzuki, 2000), irradiation creep effects are

fast neutron flux, cladding temperature and hoop stress (MATPRO-09, 1976). Thus creep strain increases as fast neutron flux, irradiation time, cladding temperature and stress increase. Fast neutron flux is predominant in cladding creep rate, whereas thermal neutron distribution is a determining factor for reactivity and thermal power (temperature of cladding) in core. It can be seen that both types of neutron flux are important for the

τ

ν

τ

and CF.

 after ν

=1 cycle/day, taking into account the estimated error < 0.4 %

*<sup>l</sup> q* =229.2 W/cm at *N*=100 %) has been

ν

ν

is expressed with a function of

=1; 2; 4 cycle/day,

had increased 4 times, in

τ

showed that if

decreases

temperature and deformation distributions physically depend on each other, simultaneous equations of thermal conduction and mechanical deformation are solved (Suzuki, 2000).

It is obvious that the cladding temperature at the central point of an AS increases when the outer oxide layer thickness increases. At the same time, according to the creep model (MATPRO-09, 1976) used in the code, the rate of equivalent creep strain max( ) *<sup>e</sup> p* τ for the central point of an axial segment increases when the corresponding cladding temperature increases. Hence the waterside corrosion of cladding is associated with the evaluation of SDE through the сreep rate depending on the thickness of metal wall (Pelykh and Maksimov, 2011).

It should be noted, that neutron irradiation has a great influence on the zircaloy corrosion behavior. Power maneuvering will alter neutron flux to give a feedback to the corrosion behavior, either positive or negative. But in this paper, the EPRI model and MATPRO code are used in the corrosion model, where irradiation term is not evidently shown. Although either temperature or reactivity coefficient is introduced in applying the model, it does not fully represent such situation.

For the studied conditions, the maximum cladding hoop stress, plastic strain and oxide layer outer surface temperature do not limit cladding durability according to the known restrictions max 250 *MPa* σθ ≤ , max θ , ε *pl* ≤ 0.5% (Novikov et al., 2005) and max *Tоx,out* ≤ 352 ºC (Shmelev et al., 2004), respectively. A similar result has been obtained for the corrosion model MATPRO-A.

Setting COR = 0 and COR = 1 (MATPRO-А), the SDE values for the algorithms 55-31-55-55 and 55-31-69-82 have been calculated. Then the numbers of calendar daily cycles prior to the beginning of rapid creep stage for Zircaloy-4 (Pelykh and Maksimov, 2011) and rapid ω τ( ) stage for E-110 alloy (Novikov et al., 2005) have been compared under WWER-1000 conditions – see Fig. 11.

Fig. 11. Cladding damage parameter (E-110) and SDE (Zircaloy-4) as functions of time: (1) ω τ( ) according to equation (2); (2.1, 2.2) *A*( ) τ at COR = 0 for the algorithms 55-31-55-55 and 55-31-69-82, respectively; (3.1, 3.2) *A*( ) τ at COR = 1 for the algorithms 55-31-55-55 and 55-31- 69-82, respectively.

temperature and deformation distributions physically depend on each other, simultaneous equations of thermal conduction and mechanical deformation are solved (Suzuki, 2000).

It is obvious that the cladding temperature at the central point of an AS increases when the outer oxide layer thickness increases. At the same time, according to the creep model

central point of an axial segment increases when the corresponding cladding temperature increases. Hence the waterside corrosion of cladding is associated with the evaluation of SDE through the сreep rate depending on the thickness of metal wall (Pelykh and

It should be noted, that neutron irradiation has a great influence on the zircaloy corrosion behavior. Power maneuvering will alter neutron flux to give a feedback to the corrosion behavior, either positive or negative. But in this paper, the EPRI model and MATPRO code are used in the corrosion model, where irradiation term is not evidently shown. Although either temperature or reactivity coefficient is introduced in applying the model, it does not

For the studied conditions, the maximum cladding hoop stress, plastic strain and oxide layer outer surface temperature do not limit cladding durability according to the known

(Shmelev et al., 2004), respectively. A similar result has been obtained for the corrosion

Setting COR = 0 and COR = 1 (MATPRO-А), the SDE values for the algorithms 55-31-55-55 and 55-31-69-82 have been calculated. Then the numbers of calendar daily cycles prior to the beginning of rapid creep stage for Zircaloy-4 (Pelykh and Maksimov, 2011) and rapid

stage for E-110 alloy (Novikov et al., 2005) have been compared under WWER-1000

Fig. 11. Cladding damage parameter (E-110) and SDE (Zircaloy-4) as functions of time: (1)

τ

τ

*pl* ≤ 0.5% (Novikov et al., 2005) and max *Tоx,out* ≤ 352 ºC

at COR = 0 for the algorithms 55-31-55-55 and

at COR = 1 for the algorithms 55-31-55-55 and 55-31-

τ

for the

ω τ( )

(MATPRO-09, 1976) used in the code, the rate of equivalent creep strain max( ) *<sup>e</sup> p*

Maksimov, 2011).

fully represent such situation.

restrictions max 250 *MPa* σθ

model MATPRO-A.

conditions – see Fig. 11.

ω τ

69-82, respectively.

≤ , max

( ) according to equation (2); (2.1, 2.2) *A*( )

55-31-69-82, respectively; (3.1, 3.2) *A*( )

θ , ε

It is necessary to notice that line 1 in Fig. 11 was calculated using separate consideration of steady-state operation and varying duty. When using equation (2), the fatigue component has an overwhelming size in comparison with the static one (Novikov et al., 2005).

Use of the MATPRO-А corrosion model under the WWER-1000 core conditions ensures conservatism of the E-110 cladding durability estimation (see Fig. 11). Growth rate of *A*(τ) depends significantly on the FA transposition algorithm. The number of daily cycles prior to the beginning of rapid creep stage decreases significantly when COR (cladding outer surface corrosion rate) increases.

Setting the WWER-1000 regime and FA constructional parameters, a calculation study of Zircaloy-4 cladding fatigue factor at variable load frequency ν << 1 Hz, under variable loading, was carried out. The investigated WWER-1000 fuel cladding had an outer diameter and thickness of 9.1 mm and 0.69 mm, respectively. The microstructure of Zircaloy-4 was a stress-relieved state. Using the cladding corrosion model EPRI (Suzuki, 2000), AS 6 of a medium-load FE in FA 55 (maximum LHR max *<sup>l</sup> q* =229.2 W/cm at *N*=100 %) has been analysed (COR = 1, inlet coolant temperature *Tin=*const=287 °C). The variable loading cycle 100–80–100 % was studied for Δτ =11; 5; 2 h (reactor capacity factor CF=0.9): *N* lowering from 100 to 80 % for 1 h → exploitation at *N* = 80 % for Δτ h → *N* rising to *N*nom=100 % for 1 h → exploitation at *N* = 100 % for Δτ h, corresponding to ν =1; 2; 4 cycle/day, respectively(ν<< 1 Hz).

Calculation of the cladding failure beginning moment τ0 depending on ν showed that if ν << 1 Hz and CF=idem, then there was no decrease of 0 τ after ν had increased 4 times, in comparison with the case ν =1 cycle/day, taking into account the estimated error < 0.4 % (η=0.4, AS 6). At the same time, when *N*=100 % =const (CF=1), the calculated 0 τ decreases significantly – see Table 10.

Hence, the WWER-1000 FE cladding durability estimation based on the CET model corresponds to the experimental results (Kim et al., 2007) in principle.


Table 10. Change of cladding failure time depending on νand CF.

In the creep model used in the FEMAXI code (Suzuki, 2000), irradiation creep effects are taken into consideration and cladding creep strain rate ( ) *<sup>e</sup> p* τ is expressed with a function of fast neutron flux, cladding temperature and hoop stress (MATPRO-09, 1976). Thus creep strain increases as fast neutron flux, irradiation time, cladding temperature and stress increase. Fast neutron flux is predominant in cladding creep rate, whereas thermal neutron distribution is a determining factor for reactivity and thermal power (temperature of cladding) in core. It can be seen that both types of neutron flux are important for the cladding life.

Theory of Fuel Life Control Methods at Nuclear

*N T*< > is expressed as

power coefficient of reactivity, respectively.

δ

In case of the assumption

equation (23) is simplified:

reactor power maneuvering.

*N*

δ

*k*

core, respectively.

The term / δ δ

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 221

average coolant temperature small deviations for the upper half-core and for the lower half-

*N Tk T Nk*

where ρ is reactivity; *Tk* and *kN* are the coolant temperature coefficient of reactivity and the

Having substituted equations (20)−(22) in (19), the following equation for a small deviation

<sup>0</sup> [(1 ) (1 ) ] *<sup>T</sup>*

<sup>0</sup> [ ] *<sup>T</sup>*

 δ*T T*

δδ

*<sup>k</sup> NT T*

The criterion of AO stabilization due to the coolant temperature coefficient of reactivity (the

min [ ]

where *i* is the power step number; *m* is the total number of power steps in some direction at

Use of the criterion (26) allows us to select a coolant temperature regime giving the maximum LHR axial distribution stability at power maneuvering. Let us study the following three WWER-1000 power maneuvering methods: M-1 is the method with a constant inlet coolant temperature *Tin*=const; M-2 is the method with a constant average coolant temperature <*T*>=const; and M-3 is the intermediate method having *Tin* increased by 1 °C only, when *N* lowers from 100% to 80%. Сomparison of these power maneuvering methods has been made using the RS code. Distribution of long-lived and stable fission products causing reactor slagging was specified for the KhNPP Unit 2 fifth campaign start, thus the first core state having an equilibrium xenon distribution was calculated at this moment. The non-equilibrium xenon and samarium distributions were calculated for subsequent states taking into account the fuel burnup. The coolant inlet pressure and coolant flow rate were specified constant and equal to 16 МPа and 84103 m3/h, respectively. When using М-1, the coolant inlet temperature was specified at *Tin*=287 °C . When using М-

δδ<sup>−</sup> = ⋅ ⋅ − ⋅ < >− + ⋅ < > *u l* (23)

АО<sup>0</sup> << 1 (24)

<sup>−</sup> = ⋅ ⋅ < >− < > *u l* (25)

< >− < > *u l* , (26)

*<sup>k</sup> NT T*

 δρ δ

АО АО0 0 АО <sup>1</sup>

 δρ δ

< > = ≡ < >

δ

of AO caused by a small deviation of *N* is derived after linearization:

АО <sup>1</sup>

*k*

coolant temperature regime effectiveness criterion) is obtained from (25):

1

δ

*i*

=

*m*

*N*

δ

/ , /

*T N* (22)

One of main tasks at power maneuvering is non-admission of axial power flux xenon waves in the active core. Therefore, for a power-cycling WWER-1000 nuclear unit, it is interesting to consider a cladding rupture life control method on the basis of stabilization of neutron flux axial distribution. The well-known WWER-1000 power control method based on keeping the average coolant temperature constant has such advantages as most favorable conditions for the primary coolant circuit equipment operation, аs well as possibility of stable NR power regulation due to the temperature coefficient of reactivity. However, this method has such defect as an essential raise of the secondary circuit steam pressure at power lowering, which requires designing of steam generators able to work at an increased pressure.

Following from this, it is an actual task to develop advanced power maneuvering methods for the ENERGOATOM WWER-1000 units which have such features as neutron field axial distribution stability, favorable operation conditions for the primary circuit equipment, especially for FE claddings, as well as avoidance of a high pressure steam generator design. The described daily power maneuvering method with a constant inlet coolant temperature allows to keep the secondary circuit initial steam pressure within the standard range of 58- 60 bar (*N*=100-80%).

The nonstationary reactor poisoning adds a positive feedback to any neutron flux deviation. Therefore, as influence of the coolant temperature coefficient of reactivity is a fast effect, while poisoning is a slow effect having the same sign as the neutron flux deviation due to this reactivity effect, and strengthening it due to the positive feedback, it can be expected that a correct selection of the coolant temperature regime ensures the neutron flux density axial distribution stability at power maneuvering. The neutron flux axial stability is characterized by AO (Philipchuk et al., 1981):

$$\text{AO} = \frac{N\_u - N\_l}{N} \,\text{\,\,\,}\tag{19}$$

where *Nu*, *Nl*, *N* are the core upper half power, lower half power and whole power, respectively.

The variables AO, *Nu*, *Nl*, *N* are represented as

$$\text{AO} = \text{AO}\_0 + \delta \text{AO} \; ; \; N\_u = N\_{u,0} + \delta N\_{u\prime} \; ; \; N\_l = N\_{l,0} + \delta N\_{l\prime} \; ; \; N = N\_0 + \delta N \; , \tag{20}$$

where АО0 ,0 ,0 0 , , , *Nu l N N* are the stationary values of АО, , , *Nu l N N* , respectively; δ δδδ АО, , , *Nu l N N* are the sufficiently small deviations from АО0 ,0 ,0 0 , , , *Nu l N N* , respectively.

The small deviations of *Nu* and *Nl* caused by the relevant average coolant temperature deviations δ < > *Tu* and δ< > *T<sup>l</sup>* are expressed as

$$
\delta \mathcal{N}\_{\mu} = \frac{\delta \mathcal{N}}{\delta < T >} \cdot \delta < T\_{\mu} >; \delta \mathcal{N}\_{l} = \frac{\delta \mathcal{N}}{\delta < T >} \cdot \delta < T\_{l} >,\tag{21}
$$

where δ δ *N N* and *u l* are the small deviations of *Nu* and *Nl* , respectively; δ < > *T* is the average coolant temperature small deviation for the whole core; δ δ< > <> *T T* and *u l* are the average coolant temperature small deviations for the upper half-core and for the lower halfcore, respectively.

The term / δ δ*N T*< > is expressed as

220 Nuclear Reactors

One of main tasks at power maneuvering is non-admission of axial power flux xenon waves in the active core. Therefore, for a power-cycling WWER-1000 nuclear unit, it is interesting to consider a cladding rupture life control method on the basis of stabilization of neutron flux axial distribution. The well-known WWER-1000 power control method based on keeping the average coolant temperature constant has such advantages as most favorable conditions for the primary coolant circuit equipment operation, аs well as possibility of stable NR power regulation due to the temperature coefficient of reactivity. However, this method has such defect as an essential raise of the secondary circuit steam pressure at power lowering, which requires designing of steam generators able to work at an increased

Following from this, it is an actual task to develop advanced power maneuvering methods for the ENERGOATOM WWER-1000 units which have such features as neutron field axial distribution stability, favorable operation conditions for the primary circuit equipment, especially for FE claddings, as well as avoidance of a high pressure steam generator design. The described daily power maneuvering method with a constant inlet coolant temperature allows to keep the secondary circuit initial steam pressure within the standard range of 58-

The nonstationary reactor poisoning adds a positive feedback to any neutron flux deviation. Therefore, as influence of the coolant temperature coefficient of reactivity is a fast effect, while poisoning is a slow effect having the same sign as the neutron flux deviation due to this reactivity effect, and strengthening it due to the positive feedback, it can be expected that a correct selection of the coolant temperature regime ensures the neutron flux density axial distribution stability at power maneuvering. The neutron flux axial stability is

> АО *N N N*

where *Nu*, *Nl*, *N* are the core upper half power, lower half power and whole power,

АО АО АО <sup>0</sup> ,0 ,0 <sup>0</sup> = + = + = + =+

 δ

where АО0 ,0 ,0 0 , , , *Nu l N N* are the stationary values of АО, , , *Nu l N N* , respectively;

The small deviations of *Nu* and *Nl* caused by the relevant average coolant temperature

; *N N N TN T T T*

= ⋅< > = ⋅< >

δδ

*N N* and *u l* are the small deviations of *Nu* and *Nl* , respectively;

 δ

 δ

< > *T<sup>l</sup>* are expressed as

δ

average coolant temperature small deviation for the whole core;

δ

АО, , , *Nu l N N* are the sufficiently small deviations from АО0 ,0 ,0 0 , , , *Nu l N N* ,

<sup>−</sup> <sup>=</sup> *u l* , (19)

δδ

δ

 δ< > <> *T T* and *u l* are the

< > *T* is the

; ; ; *N N N N N NN N N u u ul l l* , (20)

 δ

δ

< > < > *<sup>u</sup> u l <sup>l</sup>* , (21)

pressure.

60 bar (*N*=100-80%).

respectively.

deviations

where

δ

respectively.

δδδ

δ

< > *Tu* and

 δ

δ

characterized by AO (Philipchuk et al., 1981):

The variables AO, *Nu*, *Nl*, *N* are represented as

δ

δ

δ

$$\frac{\delta \mathcal{N}}{\delta < T >} = \frac{\delta \rho \;/\ \; \delta < T >}{\delta \rho \;/\ \; \delta \mathcal{N}} \equiv \frac{k\_T}{k\_N} \; \prime \tag{22}$$

where ρ is reactivity; *Tk* and *kN* are the coolant temperature coefficient of reactivity and the power coefficient of reactivity, respectively.

Having substituted equations (20)−(22) in (19), the following equation for a small deviation of AO caused by a small deviation of *N* is derived after linearization:

$$
\delta \text{AO} = \frac{k\_T}{k\_N} \cdot N\_0^{-1} \cdot \left[ (1 - \text{AO}\_0) \cdot \delta < T\_u > -(1 + \text{AO}\_0) \cdot \delta < T\_l > \right] \tag{23}
$$

In case of the assumption

$$\text{AO}\_0 \ll 1 \tag{24}$$

equation (23) is simplified:

$$
\delta \text{AOO} = \frac{k\_T}{k\_N} \cdot N\_0^{-1} \cdot \left[ \delta < T\_u > -\delta < T\_l > \right] \tag{25}
$$

The criterion of AO stabilization due to the coolant temperature coefficient of reactivity (the coolant temperature regime effectiveness criterion) is obtained from (25):

$$\min \left| \sum\_{i=1}^{m} [\delta < T\_u > -\delta < T\_l >] \right| \tag{26}$$

where *i* is the power step number; *m* is the total number of power steps in some direction at reactor power maneuvering.

Use of the criterion (26) allows us to select a coolant temperature regime giving the maximum LHR axial distribution stability at power maneuvering. Let us study the following three WWER-1000 power maneuvering methods: M-1 is the method with a constant inlet coolant temperature *Tin*=const; M-2 is the method with a constant average coolant temperature <*T*>=const; and M-3 is the intermediate method having *Tin* increased by 1 °C only, when *N* lowers from 100% to 80%. Сomparison of these power maneuvering methods has been made using the RS code. Distribution of long-lived and stable fission products causing reactor slagging was specified for the KhNPP Unit 2 fifth campaign start, thus the first core state having an equilibrium xenon distribution was calculated at this moment. The non-equilibrium xenon and samarium distributions were calculated for subsequent states taking into account the fuel burnup. The coolant inlet pressure and coolant flow rate were specified constant and equal to 16 МPа and 84103 m3/h, respectively. When using М-1, the coolant inlet temperature was specified at *Tin*=287 °C . When using М-

Theory of Fuel Life Control Methods at Nuclear

and М-3; (2) the method М-2-b.

by the RS code. Let us enter the simplifying representation

subject to time was set under the linear law (Fig. 13).

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 223

lowered from *N*2=90% to *N*3=80% within 2.5 h, under the law d*N*2-3/dτ= –0.4%/6 min, at the expense of reactor poisoning. The coolant concentration of boric acid was the criticality parameter when *N* stayed constant during 4 h. The NR power increased from *N*3=80% to *N*1=100% within 2 h, under the law d*N*3-1/dτ=1.0%/ 6 min, at the expense of pure distillate water entering and synchronous return of the regulating group to the scheduled position. When *N* increased from *N*3=80% tо *N*1=100%, change of the regulating group position *H*

Fig. 13. Change of the regulating group position subject to time: (1) the methods М-1, М-2-а

Thus, modelling of the non-equilibrium WWER-1000 control was made by assignment of the following control parameters: criticality parameter; *Tin*,0 ; d*Tin*/d*N*; *N*1; *N*2; *N*3; *H*0; Δ*H*; d*N*/dτ. Setting the WWER-1000 operation parameters in accordance with the Shmelev's method (Shmelev et al., 2004), for the methods М-1, М-2-а, М-2-b and М-3, when *N* changed from 100% to 80%, the change of core average LHR distribution was calculated

Δ ≡ < >− < >

 δ

*TT T u l* (27)

δδ


2, the coolant inlet temperature was specified according to Table 11 (*Tout* is the coolant outlet temperature).

Table 11. Change of the coolant temperature at <*T*>=const in the M-2 method.

Denoting change of the lowest control rod axial coordinate (%) measured from the core bottom during a power maneuvering as Δ*H*, the first (М-2-а) and second (М-2-b) variants of М-2 had the regulating group movement amplitudes Δ*H*2а =4% and Δ*H*2b =6%, respectively. The reactor power change subject to time was set according to the same time profile for all the methods (Fig. 12).

Fig. 12. Change of the reactor power subject to time.

For all the methods, *N* lowered from *N*1=100 % to *N*2=90 % within 0.5 h, under the linear law d*N*1-2/dτ=–2%/6 min, at the expense of boric acid entering. Also for all the methods, *N* 

2, the coolant inlet temperature was specified according to Table 11 (*Tout* is the coolant outlet

Denoting change of the lowest control rod axial coordinate (%) measured from the core bottom during a power maneuvering as Δ*H*, the first (М-2-а) and second (М-2-b) variants of М-2 had the regulating group movement amplitudes Δ*H*2а =4% and Δ*H*2b =6%, respectively. The reactor power change subject to time was set according to the same time profile for all

For all the methods, *N* lowered from *N*1=100 % to *N*2=90 % within 0.5 h, under the linear law d*N*1-2/dτ=–2%/6 min, at the expense of boric acid entering. Also for all the methods, *N* 

*N*, % *Tin*, °C *Tout*, °C <*T*>, °C 100 287 317 302 90 288 316 302 80 290 314 302

Table 11. Change of the coolant temperature at <*T*>=const in the M-2 method.

temperature).

the methods (Fig. 12).

Fig. 12. Change of the reactor power subject to time.

lowered from *N*2=90% to *N*3=80% within 2.5 h, under the law d*N*2-3/dτ= –0.4%/6 min, at the expense of reactor poisoning. The coolant concentration of boric acid was the criticality parameter when *N* stayed constant during 4 h. The NR power increased from *N*3=80% to *N*1=100% within 2 h, under the law d*N*3-1/dτ=1.0%/ 6 min, at the expense of pure distillate water entering and synchronous return of the regulating group to the scheduled position. When *N* increased from *N*3=80% tо *N*1=100%, change of the regulating group position *H* subject to time was set under the linear law (Fig. 13).

Fig. 13. Change of the regulating group position subject to time: (1) the methods М-1, М-2-а and М-3; (2) the method М-2-b.

Thus, modelling of the non-equilibrium WWER-1000 control was made by assignment of the following control parameters: criticality parameter; *Tin*,0 ; d*Tin*/d*N*; *N*1; *N*2; *N*3; *H*0; Δ*H*; d*N*/dτ. Setting the WWER-1000 operation parameters in accordance with the Shmelev's method (Shmelev et al., 2004), for the methods М-1, М-2-а, М-2-b and М-3, when *N* changed from 100% to 80%, the change of core average LHR distribution was calculated by the RS code. Let us enter the simplifying representation

$$
\Delta \delta T \equiv \delta < T\_u > -\delta < T\_l > \tag{27}
$$

Theory of Fuel Life Control Methods at Nuclear

are identical.

method, the value of

η

η

 τ*<sup>e</sup>* = ; lim *n n p p* < ; max

0 00 () ()

moment 0 τ

the choice of

value of

 τ ησ

σ

η

, when the condition max max

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 225

Fig. 14. Equilibrium and instant axial offsets (1, 2a, 2b, 3) subject to time for М-1, М-2-а, М-2 b and М-3, respectively: (lower line) the equilibrium АО; (upper line) the instant АО.

The regulating group movement amplitude is the same (4%) for M-1, М-2-а and М-3, but the maximum divergence between the instant and equilibrium offsets are max ΔAO1 ≈ 1.9% (M-1), max ΔAO2a ≈ 3% (M-2-a) and max ΔAO3 ≈ 2.3% (M-3). This result confirms the conclusion made on the basis of the criterion (26). If the regulating group movement amplitude, at power maneuvering according to the method with a constant average coolant temperature, is increased from 4 tо 6%, then the maximum AO divergence lowers from 3% tо 1.9% (see Fig. 14). Therefore, when using the method with < > *T* =const, a greater regulating group movement amplitude is needed to guarantee the LHR axial stability, than when using the method with *Tin* = const, on the assumption that all other conditions for both the methods

Having used the RS code, the core average LHR axial distribution change has been calculated for the methods М-1, М-2-а, М-2-b and М-3, for the following daily power maneuvering cycle: lowering of *N* from *N*1=100% to *N*2=90% during 0.5 h by injection of boric acid solution – further lowering of *N* to *N*3=80% during 2.5 h due to reactor poisoning – operation at *N*3=80% during 4 h – rising of *N* tо the nominal capacity level *N*1=100% during 2 h at the expense of pure distillate water entering and synchronous return of the

When using the criterion (15) for comparative analysis of cladding durability subject to the FA transposition algorithm, the position of an axial segment and the power maneuvering

maximum number of power history points is limited by lim *np* =10,000 in the FEMAXI code,

maneuvering method, because a greater time period as well as a more complicated power maneuvering method are described by a greater number of history points *np* . Therefore, the

should be specified on the basis of simultaneous conditions max max

0 00 () ()

should be set taking into account the necessity of determining the

τ

*<sup>e</sup>* = is satisfied. In addition, as the

≤ . Though the cladding failure parameter values

and the complexity of a power

 τ

regulating group to the scheduled position – operation at *N*1=100% during 15 h.

σ

depends on the analysed time period max

0 τ τ  τ ησ

Using the obtained LHR distribution, by the FEMAXI code (Suzuki, 2010), the average coolant temperatures of the upper < > *Tu* and lower < > *Tl* half-cores were calculated for М-1, М-2-а and М-3 (at time step 0.5 h). Then, on the basis of < > *Tu* and < > *T<sup>l</sup>* , 6 *i* 1 δ*T* = <sup>Δ</sup> was


found for М-1, М-2-а and М-3 having the same Δ*H* (Table 12).

Table 12. Change of the average coolant temperatures for М-1, М-2-а, М-3.

Having used the criterion (26), the conclusion follows that the coolant temperature regime М-1 ensures the most stable АО, while the regime М-2-а is least favorable − see Table 12. In order to check this conclusion, it is useful to compare АО stabilization for the discussed methods, calculating the divergence ΔAO between the instant and equilibrium axial offsets (Philimonov and Мамichev, 1998) − see Fig. 14.

Using the obtained LHR distribution, by the FEMAXI code (Suzuki, 2010), the average coolant temperatures of the upper < > *Tu* and lower < > *Tl* half-cores were calculated for М-

> δ< > *T<sup>u</sup>*

0.6 90 317.975 296.575 -0.325 -0.25 -0.075 1.1 88 316.375 296 -1.6 -0.575 -1.025 1.6 86 315.725 295.725 -0.65 -0.275 -0.375 2.1 84 315.1 295.525 -0.625 -0.2 -0.425 2.6 82 314.5 295.3 -0.6 -0.225 -0.375 3.1 80 313.9 295.075 -0.6 -0.225 -0.375

0.6 90 319.25 298.025 0.95 1.2 -0.25

1.1 88 317.875 297.575 -1.375 -0.45 -0.925 1.6 86 317.45 297.575 -0.425 0 -0.425 2.1 84 316.925 297.575 -0.525 0 -0.525 2.6 82 316.65 297.625 -0.275 0.05 -0.325 3.1 80 316.35 297.725 -0.3 0.1 -0.4

0.6 90 318.4 297.075 0.1 0.25 -0.15

1.1 88 316.9 296.55 -1.5 -0.525 -0.975 1.6 86 316.35 296.4 -0.55 -0.15 -0.4 2.1 84 315.7 296.2 -0.65 -0.2 -0.45 2.6 82 315.225 296.125 -0.475 -0.075 -0.4 3.1 80 314.775 296 -0.45 -0.125 -0.325

Having used the criterion (26), the conclusion follows that the coolant temperature regime М-1 ensures the most stable АО, while the regime М-2-а is least favorable − see Table 12. In order to check this conclusion, it is useful to compare АО stabilization for the discussed methods, calculating the divergence ΔAO between the instant and equilibrium axial offsets

Table 12. Change of the average coolant temperatures for М-1, М-2-а, М-3.

(Philimonov and Мамichev, 1998) − see Fig. 14.

δ

< > *T<sup>l</sup>* Δ

6

<sup>Δ</sup> was

6

*i* 1 δ*T*

2.65

2.85

2.70

= <sup>Δ</sup>

*i* 1 δ*T*

=

δ*T*

1, М-2-а and М-3 (at time step 0.5 h). Then, on the basis of < > *Tu* and < > *T<sup>l</sup>* ,

M-1; M-2-а; M-3 0.1 100 318.3 296.825 0 0 0

found for М-1, М-2-а and М-3 having the same Δ*H* (Table 12).

Мethod τ, h *N,* % < > *T<sup>u</sup>* < > *T<sup>l</sup>*

M-1

M-2-а

M-3

Fig. 14. Equilibrium and instant axial offsets (1, 2a, 2b, 3) subject to time for М-1, М-2-а, М-2 b and М-3, respectively: (lower line) the equilibrium АО; (upper line) the instant АО.

The regulating group movement amplitude is the same (4%) for M-1, М-2-а and М-3, but the maximum divergence between the instant and equilibrium offsets are max ΔAO1 ≈ 1.9% (M-1), max ΔAO2a ≈ 3% (M-2-a) and max ΔAO3 ≈ 2.3% (M-3). This result confirms the conclusion made on the basis of the criterion (26). If the regulating group movement amplitude, at power maneuvering according to the method with a constant average coolant temperature, is increased from 4 tо 6%, then the maximum AO divergence lowers from 3% tо 1.9% (see Fig. 14). Therefore, when using the method with < > *T* =const, a greater regulating group movement amplitude is needed to guarantee the LHR axial stability, than when using the method with *Tin* = const, on the assumption that all other conditions for both the methods are identical.

Having used the RS code, the core average LHR axial distribution change has been calculated for the methods М-1, М-2-а, М-2-b and М-3, for the following daily power maneuvering cycle: lowering of *N* from *N*1=100% to *N*2=90% during 0.5 h by injection of boric acid solution – further lowering of *N* to *N*3=80% during 2.5 h due to reactor poisoning – operation at *N*3=80% during 4 h – rising of *N* tо the nominal capacity level *N*1=100% during 2 h at the expense of pure distillate water entering and synchronous return of the regulating group to the scheduled position – operation at *N*1=100% during 15 h.

When using the criterion (15) for comparative analysis of cladding durability subject to the FA transposition algorithm, the position of an axial segment and the power maneuvering method, the value of η should be set taking into account the necessity of determining the moment 0 τ , when the condition max max 0 00 () () σ τ ησ τ *<sup>e</sup>* = is satisfied. In addition, as the maximum number of power history points is limited by lim *np* =10,000 in the FEMAXI code, the choice of η depends on the analysed time period max τ and the complexity of a power maneuvering method, because a greater time period as well as a more complicated power maneuvering method are described by a greater number of history points *np* . Therefore, the value of η should be specified on the basis of simultaneous conditions max max 0 00 () () σ τ ησ τ *<sup>e</sup>* = ; lim *n n p p* < ; max 0 τ τ≤ . Though the cladding failure parameter values

Theory of Fuel Life Control Methods at Nuclear

max σ

fuel burnup compromise.

fuel assemblies.

this physically based method.

Power Plants (NPP) with Water-Water Energetic Reactor (WWER) 227

FA transposition algorithm 55-31-69-82 55-31-55-55

*BU*, МW·day/kg 57.4 71.4

Thus, an optimal FA transposition algorithm must be set on the basis of cladding durability-

As is shown, the operating reactor power history as well as the WWER–1000 main regime and design parameters included into the second conditional group (pellet hole diameter, cladding thickness, pellet effective density, maximum FE linear heat rate, etc.) influence significantly on fuel cladding durability. At normal operation conditions, the WWER-1000 cladding corrosion rate is determined by design constraints for cladding and coolant, and depends slightly on the regime of variable loading. Also the WWER-1000 FE cladding rupture life, at normal variable loading operation conditions, depends greatly on the coolant temperature regime and the FA transposition algorithm. In addition, choice of the group of regulating units being used at NR power maneuvering influences greatly on the offset

Hence, under normal operation conditions, the following methods of fuel cladding


To create a computer-based fuel life control system at NPP with WWER, it is necessary to calculate the nominal and maximum permissible values of pick-off signals on the basis of calculated FA normal operation probability (Philipchuk et al., 1981). Though a computerbased control system SAKOR-M has already been developed for NPP with WWER at the OKB "Gidropress" (Bogachev et al., 2007), this system does not control the remaining life of

As the described CET-method can be applied to any type of LWR including prospective thorium reactors, the future fuel life control system for NPP with LWR can be created using

Table 14. Fuel burnup and cladding equivalent creep strain for AS 6 (after 1500 d).

**5. Methods of fuel cladding durability control at NPP with WWER** 

stabilization efficiency (Philimonov and Мамichev, 1998).


making the fuel pellets with centre holes; - assignment of the coolant temperature regime; - assignment of the FA transposition algorithm;

durability control at NPP with WWER can be considered as main ones:


COR 0 1 0 1

*<sup>e</sup>* , МPa (% of σ0) 69.9 (33) 127.4 (61) 107.2 (51) 146.7 (70)

*pe*, % 4.22 11.22 9.36 16.02

listed in Table 6 were obtained assuming η =0.6 (the MATPRO-A corrosion model, COR = 1), comparison of cladding failure parameters for different power maneuvering methods can be made using the cladding collapse criterion (15), for instance, at η =0.4. Assuming η =0.4, on the basis of the obtained LHR distributions, the cladding failure parameters have been calculated by the instrumentality of the FEMAXI code (Suzuki, 2000) for the methods М-1, М-2-а, М-2-b and М-3, for the axial segments six and seven (the MATPRO-A corrosion model, COR = 1) – see Table 13.


Table 13. Cladding failure parameters for the methods М-1, М-2-а, М-2-b and М-3.

Among the regimes with the regulating group movement amplitude Δ*H* =4%, the coolant temperature regime М-1 ensuring the most stable АО is also characterized by the least calculated cladding failure parameter ω(500 ) days , while the regime М-2-a having the least stable АО is also characterized by the greatest ω(500 ) days – see Table 12, Fig. 14 and Table 13. The intermediate method M-3 having *Tin* increased by 1 °С only, when *N* lowers from 100% to 80%, is also characterized by the intermediate values of AO stability and ω(500 ) days .

In addition, the second variant of М-2 (М-2-b) having the regulating group movement amplitude Δ*H*2b =6% is characterized by a more stable AO in comparison with the method М-2-a (see Fig. 14) and, for the most strained axial segment six, by a greater value of ω(500 ) days – see Table 13.

It should be stressed that the proposed cladding rupture life control methods are not limited only in WWER-1000. Using the FEMAXI code, these methods can be extended into other reactor types (like PWR or BWR). At the same time, taking into account a real disposition of regulating units, a real coolant temperature regime as well as a real FA transposition algorithm, in order to estimate the amplitude of LHR jumps at FE axial segments occurring when the NR (PWR or BWR) capacity periodically increases, it is necessary to use another code instead of the RS code, which was developed for the WWER-1000 reactors.

The FA transposition algorithm 55-31-69-82 is characterized by a lower fuel cladding equivalent creep strain than the algorithm 55-31-55-55. At the same time, it has a lower fuel burnup than the algorithm 55-31-55-55 (see Table 14).

1), comparison of cladding failure parameters for different power maneuvering methods can

on the basis of the obtained LHR distributions, the cladding failure parameters have been calculated by the instrumentality of the FEMAXI code (Suzuki, 2000) for the methods М-1, М-2-а, М-2-b and М-3, for the axial segments six and seven (the MATPRO-A corrosion

Меthod М-1 М-2-а М-2-b М-3

, 504.4 497.4 496.0 501.4

<sup>0</sup> *A* , 1.061 1.094 1.080 1.068

, 530.0 519.4 519.0 525.0

<sup>0</sup> *<sup>A</sup>* , 1.044 1.055 1.019 1.043

(500 ) days 0.957 1.027 (+7.3%) 1.040 (+ 8.7%) 0.988 (+3.2%)

(500 ) days 0.766 0.848 (+10.7%) 0.848 (+10.7%) 0.804 (+5.0%)

(500 ) days , while the regime М-2-a having the least

(500 ) days – see Table 12, Fig. 14 and Table

=0.6 (the MATPRO-A corrosion model, COR =

η

=0.4. Assuming

η=0.4,

η

Table 13. Cladding failure parameters for the methods М-1, М-2-а, М-2-b and М-3.

ω

code instead of the RS code, which was developed for the WWER-1000 reactors.

Among the regimes with the regulating group movement amplitude Δ*H* =4%, the coolant temperature regime М-1 ensuring the most stable АО is also characterized by the least

13. The intermediate method M-3 having *Tin* increased by 1 °С only, when *N* lowers from 100% to 80%, is also characterized by the intermediate values of AO stability and

In addition, the second variant of М-2 (М-2-b) having the regulating group movement amplitude Δ*H*2b =6% is characterized by a more stable AO in comparison with the method М-2-a (see Fig. 14) and, for the most strained axial segment six, by a greater value of

It should be stressed that the proposed cladding rupture life control methods are not limited only in WWER-1000. Using the FEMAXI code, these methods can be extended into other reactor types (like PWR or BWR). At the same time, taking into account a real disposition of regulating units, a real coolant temperature regime as well as a real FA transposition algorithm, in order to estimate the amplitude of LHR jumps at FE axial segments occurring when the NR (PWR or BWR) capacity periodically increases, it is necessary to use another

The FA transposition algorithm 55-31-69-82 is characterized by a lower fuel cladding equivalent creep strain than the algorithm 55-31-55-55. At the same time, it has a lower fuel

ω

be made using the cladding collapse criterion (15), for instance, at

days <sup>0</sup> τ

days <sup>0</sup> τ

МJ/m<sup>3</sup>

МJ/m<sup>3</sup>

listed in Table 6 were obtained assuming

model, COR = 1) – see Table 13.

6

7

calculated cladding failure parameter

ω

ω

stable АО is also characterized by the greatest

burnup than the algorithm 55-31-55-55 (see Table 14).

Axial Segment

ω

ω

(500 ) days .

(500 ) days – see Table 13.


Table 14. Fuel burnup and cladding equivalent creep strain for AS 6 (after 1500 d).

Thus, an optimal FA transposition algorithm must be set on the basis of cladding durabilityfuel burnup compromise.
