Nuclear fuels can be classified into two main categories; metallic fuels and ceramic fuels. The most common metallic fuels include uranium, plutonium, and thorium (Kirillov et al.,

2007) . The advantage of metallic fuels is their high thermal conductivity; however, they suffer from low melting points and also that the fuel undergoes phase change. The three phases in a metallic uranium fuel includes a-, p~, and Y-phase. A phase changes to another phase as a function of temperature, resulting in a volume change in the fuel. In addition, metallic fuels undergo oxidation when exposed to air or water. For use in high-temperature applications, a potential fuel must have a high melting point, high thermal conductivity, and good irradiation and mechanical stability (Ma, 1983). These requirements eliminate various nuclear fuels categorized under the metallic fuels mainly due to their low melting points and high irradiation creep and swelling rates (Ma, 1983). On the other hand, ceramic fuels have promising properties, which make these fuels suitable candidates for SCWR applications. Table 2 provides basic properties of selected fuels at 0.1 MPa and 25°C (Chirkin, 1968; IAEA, 2008; Frost, 1963; Cox and Cronenberg, 1977; Leitnaker and Godfrey, 1967; Lundberg and Hobbins, 1992).

In general, ceramic fuels have good dimensional and radiation stability and are chemically compatible with most coolants and sheath materials. Consequently, this section focuses only on ceramic fuels. The ceramic fuels examined in this chapter are U02, MOX, Th02, UC, UN, UC>2-SiC, U02-C, and U02-Be0. Further, these ceramic fuels can be classified into three categories: 1) low thermal-conductivity fuels, 2) enhanced thermal-conductivity fuels, and 3) high thermal-conductivity fuels. Low thermal-conductivity fuels are U02, MOX, and Th02. Enhanced thermal-conductivity fuels are U02-SiC, U02-C, and U02-Be0; and high thermal — conductivity fuels are UC and UN.

Property	Unit	UO2	MOX	ThO2	UC	UN
Molecular Mass	amu	270.3	271.2	264	250.04	252.03
Theoretical density	kg/m[3] [4]	10960	11,074	10,000	136302	14420
Melting Point	°C	2847+30	2750	3227+150	25073 2520 2532[5]	2850+30[6]
Heat Capacity	J/kgK	235	240	235	203[7]	190
Heat of Vaporization	kJ/kg	1530	1498	—	2120	1144[8] 3325[9]
Thermal Conductivity	W/mK	8.7	7.8	9.7	21.2	14.6
Linear Expansion Coefficient	1/K	9.75×10~6	9.43×10~6	8.9[10]x10~6	10.1×10~6	7.52×10^6
Crystal Structure	—	FCC[11]	FCC	FCC	FCC	FCC

Table 2. Basic properties of selected fuels at 0.1 MPa and 25°C.

In addition to the melting point of a fuel, the thermal conductivity of the fuel is a critical property that affects the operating temperature of the fuel under specific conditions. U02 has been used as the fuel of choice in BWRs, PWRs, and CANDU reactors. The thermal conductivity of U02 is between 2 and 3 W/m K within the operating temperature range of SCWRs. 0n the other hand, fuels such as UC and UN have significantly higher thermal conductivities compared to that of U02 as shown in Fig. 9 (Cox and Cronenberg, 1977; Frost et al., 1963; IAEA, 2008; Ishimoto et al., 1995; Leitnaker and Godfrey, 1967; Khan et al., 2010, Kirillov et al., 2007; Lundberg and Hobbins, 1992; Solomon et al., 2005). Thus, under the same operating conditions, the fuel centerline temperature of high thermal conductivity fuels should be lower than that of U02 fuel.

Fig. 9. Thermal conductivities of several fuels.

Using the CET cladding durability estimation method, an analysis of the cladding (stress relieved zircaloy) durability estimation sensitivity to the WWER-1000 main regime and design initial data uncertainty, under variable loading conditions, has been done. The WWER-1000 main regime and design parameters have been devided into two groups: the parameters that influence the cladding failure conditions slightly and the parameters that determine the cladding failure conditions. The second group includes such initial parameters that any one of them gives a change of t0 estimation near 2 % (or greater) if the initial parameter has been specified at the value assignment interval of 3 %. This group consists of outer cladding diameter, pellet diameter, pellet hole diameter, cladding thickness, pellet effective density, maximum FE linear heat rate, coolant inlet temperature, coolant inlet pressure, coolant velocity, initial He pressure, FE grid spacing, etc. (Maksimov and Pelykh, 2009). For example, dependence of cladding SDE on the number of effective days N, for pellet centre hole diameter dhde = 0.140 cm, 0.112 cm and 0.168 cm, is shown in Fig. 4.

Fig. 4. Dependence of SDE on N for dhole: 0.140 cm (1); 0.112 cm (2); 0.168 cm (3).

Dependence of cladding equivalent stress (r’max (T and yield stress a’0nax (T, for the cladding point having the maximum temperature, on the number of effective days N, for dhole = 0.112 cm and 0.168 cm, is shown in Fig. 5.

Fig. 5. Dependence of cladding yield stress (1) and equivalent stress (2; 3) on N for dhoie: 0.112 cm (2); 0.168 cm (3). Determination of t0 for dhoie = 0.112 cm.

Using the value of T0 and the calculated dependence of SDE on N, the value of A0 is found — see Fig. 6.

Fig. 6. Calculation of A0.

For the combined variable load cycle, dependence of cladding SDE on the number of effective days N for a medium-loading FE of UTVS, TVS-А and TVS-W, is shown in Fig. 7.

Fig. 7. Dependence of SDE on N for UTVS, TVS-A and TVS-W.

For the combined cycle, the maximum SDE value was obtained for a medium-loading FE of the FA produced by WESTINGHOUSE, which has no pellet centre hole (see Table 1). The same result was obtained for the stationary regime of WWER-1000 (Maksimov and Pelykh,

2010) .

It has been found that cladding running time, expressed in cycles, for the WWER-1000 combined load cycle decreases from 1925 to 1351 cycles, when FE maximum LHR ql/max increases from 248 W/cm to 298 W/cm (Maksimov and Pelykh, 2010). Having done estimation of cladding material failure parameter со after 1576 ef. days, it was found that the WWER-1000 combined load cycle has an advantage in comparison with stationary operation at 100 % power level when ql/max < 273 W / cm — see Table 4.

According to FEM, a FE length is divided into n equal length AS. In the first publications devoted to the CET-method it was supposed that the central AS is most strained and shortest-lived. However, this assumption does not consider that segments differ in LHR jump value. In addition, it was assumed that a FA stays in the same place over the whole fuel operating period (Maksimov and Pelykh, 2009).

		FE maximum LHR, W/ cm
248	258 263	273	298
		Average fast neutron flux density, cm-2 s	1
	11014	1.04 1014 1.06 1014	1.11014	1.2 1014
		Stationary loading
T0, ef. d.	2211	2078 2016	1904	1631
A0, MJ/m3	33.37	35.66 36.87	39.74	47.64
о, %	60	65 68	74	94
Combined variable loading
T0, ef. d.	2246	2102 2032	1903	1576
A0, MJ/m3	27.36	29.14 30.05	32.10	37.69
о, %	57	64 67	74	100

Table 4. Cladding damage parameter for stationary loading and the combined variable loading of WWER-1000.

At last, influence of cladding corrosion rate on cladding durability at variable loading was not taken into account. Thus it is necessary to estimate influence of varying duty on all AS, to take account of a real FA transposition algorithm as well as to consider influence of cladding corrosion rate on its durability.

3. 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 MaMichev, 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.

The amplitude of LHR jumps was calculated for the following daily power maneuvering method: lowering of N from Ni=100% to N2=90% by injection of boric acid solution within 0.5 h — further lowering of N to N3=80% due to reactor poisoning within 2.5 h — operation at N3=80% within 4 h — rising of N to the nominal capacity level N1=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 Ni=100% and N3=80% were designated Hi=90% and H3=84%, respectively. That is when N changes from N1=100% to N3=80%, the lowest control rod axial coordinate measured from the core bottom changes from H1=90% to H3=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% to 100% level — see Table 5 (Pelykh et al., 2010).

FA group	The number of fuel assemblies	FA power growth, %	FA numbers (according to the core cartogram )
1	6	28	31, 52, 58, 106, 112, 133
2	37	26	20, 42, 43, 46, 51, 53.. .57, 66.. .71, 80.. .84, 93.. .98, 107…111, 113, 118, 121, 122, 144
3	120	< 25	all other fuel assemblies

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 q, max = 236.8 W/ cm, FA group 2) position for the first year — then the FA stays in the 31-st FA (q;max = 250.3 W/cm, group 1) position for the second year — further the FA stays in the 69-th FA (q, max = 171.9 W/cm, group 2) position for the third year — at last, the FA stays in the central 82-d FA (q, max = 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 < ql r j >. For all segments (n = 8) of the 55-th, 31-st, 69-th and 82-nd fuel assemblies, the values of < ql r j > have been calculated at power levels of N3=80% and N1=100% using the RS code. The < ql r j > (100%)/ < ql r j > (80%) ratio values are listed in Table 6.

FA number
AS	55	31	69	82
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 < ql r	j > (100%)/ < q,,,	О A	ratio values for fuel assemblies 55, 31, 69, 82.

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):

T

air) = A(T) / A0 = 1; A(r) = Jamax (r) pemax (r) dr; A0 at amax (r0) = n ОТ* (T), (15)

0

where i(r) is cladding material failure parameter; т is time, s; A(r) is SDE, J/m3; A0 is SDE at the moment r0 of cladding material failure beginning, when аЄпах(т0) = n a(5nax(T)); C^sx(t) and pm^ (r) are equivalent stress (Pa) and rate of equivalent creep strain (s-1) for the cladding point of an AS having the maximum temperature, respectively; сг™3* (T is yield stress for the cladding point of an AS having the maximum temperature, Pa; n is some factor, n < 1.

Assuming the 55-31-69-82 four-year FA transposition algorithm and n = 0.6, the i(r) values have been calculated by Eq. (15) using the following procedure: calculating aemax(r), pemax(r) and оП^Г) by the instrumentality of FEMAXI-V code (Suzuki, 2000); calculating A(t) ; determining the moment t0 according to the condition

a-max^) = n оП’Ч’Т)); determining A0 = A(t0) ; calculating i(r) — see Table 7 (Pelykh and

Maksimov, 2011).

т, days		AS
4	5	6	7
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

Table 7. Cladding failure parameters i(r) 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 i(r) value was less than 1.0, i. e. there was no cladding collapse up to т = 2495 days. For т > 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 т = 2495 days with i(r) = 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, then it stayed in the 31-st FA position for the remaining fuel operation period, the а(т) value reached 1.0 and the cladding collapse was predicted at т < 2495 days with n = 1.

The prediction shown in Table 7 that the largest value of (o(t) 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 four — year 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 / dt = (A / S2) exp(-Q1 / R Тъ )(1 + COR), (16)

where dS / dt is the oxide growth rate, pm/day; A = 6.3*109 pm3/day; S is the oxide layer thickness, pm; Ql=32289 cal/mol; R=1.987 cal/(mol K); Тъ 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

о о 1/3

S = (4.976x 10-3 A t exp(-15660 / Тъ) + S03) (1 + COR), (17)

where S is the oxide layer thickness, m; A = 1.5 (PWR); t is time, days; Тъ is the temperature at the oxide layer-metal phase boundary, K; S0 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 EPRI — model 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 pm, the maximum oxide layer thickness was restricted by 100 pm, 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.

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 T^^Ut 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 Tox Out subject to time т 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 T0XmOut =352 °C (Shmelev et al., 2004).

i	T max °c ox ,out		S ^m ( Tox	out, °C )
		360 days	720 days	1080 days	1440 days
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)

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 COR = — 0.431; 0; 1; and 2. As a result, the rapid creep stage is degenerated for both the corrosion models at COR = — 0.431 (Fig. 10).

Let us introduce a dimensionless parameter I

10-6 T

1 = °С ■ day J Tdad’in ‘dt, (18)

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

Having analysed the described method of daily power maneuvering, the maximum cladding oxide layer outer surface temperature during the period of 2400 days, as well as I(2400

days) and the 2400 days period averaged cladding inner surface temperature < Tdad in > have been calculated for the sixth segment, using the EPRI corrosion model — see Table 9.

COR	T max °c *0X	I(2400 days)	О о л "чГ h? V
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

Table 9. Cladding 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 Tciadrin (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
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 p max(T) 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 creep 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 a’m’ax < 250 MPa, ^ < 0.5% (Novikov et al., 2005) and T^, < 352 °C (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 w(t) 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) co(t) according to equation (2); (2.1, 2.2) A(t) at COR = 0 for the algorithms 55-31-55-55 and 55-31-69-82, respectively; (3.1, 3.2) A(t) at COR = 1 for the algorithms 55-31-55-55 and 55-31­69-82, respectively.

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-A corrosion model under the WWER-1000 core conditions ensures conservatism of the E-110 cladding durability estimation (see Fig. 11). Growth rate of A(x) 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 v << 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 q;max =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 Ax =11; 5; 2 h (reactor capacity factor CF=0.9): N lowering from 100 to 80 % for 1 h ^ exploitation at N = 80 % for Ax h ^ N rising to Nnom=100 % for 1 h ^ exploitation at N = 100 % for Ax h, corresponding to v =1; 2; 4 cycle/day, respectively( v << 1 Hz).

Calculation of the cladding failure beginning moment x0 depending on v showed that if v << 1 Hz and CF=idem, then there was no decrease of t0 after v had increased 4 times, in comparison with the case v =1 cycle/day, taking into account the estimated error < 0.4 % (n=0.4, AS 6). At the same time, when N=100 % =const (CF=1), the calculated t0 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.

CF		0.9		1
v, cycle/day	1	2	4	—
T0, day	547.6	547.0	549.0	436.6

Table 10. Change of cladding failure time depending on v and CF.

In the creep model used in the FEMAXI code (Suzuki, 2000), irradiation creep effects are taken into consideration and cladding creep strain rate pe (t) 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.

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, as 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 58­60 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):

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

The variables AO, Nu, Ni, N are represented as

where A°o, Nu0, Nl,0- N0 are the stationary values of АО, NU, Nt, N, respectively; 8АО, SNU, 8Nl, 8N are the sufficiently small deviations from АО0, Nu0 , Nl0, N0, respectively.

The small deviations of NU and Nt caused by the relevant average coolant temperature deviations 8 < TU > and 8 < T, > are expressed as

where 8NU and 8Nt are the small deviations of NU and Nt, respectively; 8 < T > is the average coolant temperature small deviation for the whole core; 8 < TU > and 8 < Tt > are the
average coolant temperature small deviations for the upper half-core and for the lower half­core, respectively.

The term SN / S < T > is expressed as

SN _Sp/S< T > ^ fcL

S< T > Sp/SN kN’ (

where p is reactivity; kT 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:

SO _ jL. n-1 ■ [(1 — АО0) S< Tu > -(1 + АО0) S < T, >] (23)

kN

In case of the assumption

АО0 << 1

equation (23) is simplified:

SAO _. N-1. [S< Tu >-S< T, >] (25)

kN

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

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%. Comparison 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 MPa and 84 TO3 m3/h, respectively. When using M-1, the coolant inlet temperature was specified at Tin=287 °C. When using M-

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

N, %	o о	Toub °С	<T>, °С
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.

Denoting change of the lowest control rod axial coordinate (%) measured from the core bottom during a power maneuvering as AH, the first (М-2-а) and second (М-2-b) variants of М-2 had the regulating group movement amplitudes AH2a =4% and AH2b =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 Ni=100 % to N2=90 % within 0.5 h, under the linear law dN1_2/dr=-2%/6 min, at the expense of boric acid entering. Also for all the methods, N

lowered from N2=90% to N3=80% within 2.5 h, under the law dN2-3/dx= -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 N3=80% to N1=100% within 2 h, under the law dN3-1/dx=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 N3=80% to N1=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; T^ci ; dTn/dN; N1; N2; N3; H0; AH;dN/dx. 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

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

6

was

i=1

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

Method	t, h	N, %	< Tu >	< T >	S< Tu >	S<T >	AST	6 Z AST i=1
M-1; M-2-a; M-3	0.1	100	318.3	296.825	0	0	0
	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
M-1	1.6	86	315.725	295.725	-0.65	-0.275	-0.375	2.65
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
M-2-a	1.6	86	317.45	297.575	-0.425	0	-0.425
2.1	84	316.925	297.575	-0.525	0	-0.525	2.85
	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
M-3	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

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 AAO between the instant and equilibrium axial offsets (Philimonov and MaMichev, 1998) — see Fig. 14.

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 AAOmax ~ 1.9% (M-1), AAO2maax ~ 3% (M-2-a) and AAO^ax ~ 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 to 6%, then the maximum AO divergence lowers from 3% to 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 N1=100% to N2=90% during 0.5 h by injection of boric acid solution — further lowering of N to N3=80% during 2.5 h due to reactor poisoning — operation at N3=80% during 4 h — rising of N to the nominal capacity level N1=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 N1=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 n should be set taking into account the necessity of determining the moment t0, when the condition cr^To) = П C^To) is satisfied. In addition, as the maximum number of power history points is limited by npim =10,000 in the FEMAXI code, the choice of n depends on the analysed time period Tmax 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 n should be specified on the basis of simultaneous conditions CmaxT0) = П c0max(T0); np < npim; t0 < Tmax. Though the cladding failure parameter values listed in Table 6 were obtained assuming n =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 n =0.4. Assuming n =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.

	Method	М-1	М-2-а	М-2-b	M-3
		t0 , days	504.4	497.4	496.0	501.4
	6	A0, MJ/m3	1.061	1.094	1.080	1.068
Axial		ю(500 days)	0.957	1.027 (+7.3%)	1.040 (+ 8.7%)	0.988 (+3.2%)
Segment		t0 , days	530.0	519.4	519.0	525.0
	7	A0, MJ/m3	1.044	1.055	1.019	1.043
		ю(500 days)	0.766	0.848 (+10.7%)	0.848 (+10.7%)	0.804 (+5.0%)

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 AH =4%, the coolant temperature regime М-1 ensuring the most stable АО is also characterized by the least calculated cladding failure parameter r»(500 days), while the regime М-2-a having the least stable АО is also characterized by the greatest r»(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 r»(500 days).

In addition, the second variant of М-2 (М-2-b) having the regulating group movement amplitude AH2b =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 a)(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).

FA transposition algorithm	55-31-69-82	55-31-55-55
COR	0	1	0	1
Bu, MW-day/kg	57.4	71.4
CTemax, MPa (% of 00)	69.9 (33)	127.4 (61)	107.2 (51)	146.7 (70)
pe, %	4.22	11.22	9.36	16.02

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 durability — fuel burnup compromise.

Yuri Orechwa1[13] and Mihaly Makai2 1NRC, Washington DC 2BME Institute of Nuclear Techniques, Budapest

1USA

2Hungary

1. Introduction

Group theory is a vast mathematical discipline that has found applications in most of physical science and particularly in physics and chemistry. We introduce a few of the basic concepts and tools that have been found to be useful in some nuclear engineering problems. In particular those problems that exhibit some symmetry in the form of material distribution and boundaries. We present the material on a very elementary level; an undergraduate student well versed in harmonic analysis of boundary value problems should be able to easily grasp and appreciate the central concepts.

The application of group theory to the solution of physical problems has had a curious history. In the first half of the 20th century it has been called by some the "Gruppen Pest" , while others embraced it and went on to win Noble prizes. This dichotomy in attitudes to a formal method for the solution of physical problems is possible in light of the fact that the results obtained with the application of group theory can also be obtained by standard methods. In the second half of the 20th century, however, it has been shown that the formal application of symmetry and invariance through group theory leads in complicated problems not only to deeper physical insight but also is a powerful tool in simplifying some solution methods.

In this chapter we present the essential group theoretic elements in the context of crystallographic point groups. Furthermore we present only a very small subset of group theory that generally forms the first third of the texts on group theory and its physical applications. In this way we hope, in short order, to answer some of the basic questions the reader might have with regard to the mechanical aspects of the application of group theory, in particular to the solution of boundary value problems in nuclear engineering, and the benefits that can accrue through its formal application. This we hope will stimulate the reader to look more deeply into the subject is some of the myriad of available texts.

The main illustration of the application of group theory to Nuclear Engineering is presented in Section 4 of this chapter through the development of an algorithm for the solution of the neutron diffusion equation. This problem has been central to Nuclear Engineering from the very beginning, and is thereby a useful platform for demonstrating the mechanics of bringing group theoretic information to bear. The benefits of group theory in Nuclear Engineering are

not restricted to solving the diffusion equation. We wish to also point the interested reader to other areas of Nuclear Engineering were group theory has proven useful.

An early application of group theory to Nuclear Engineering has been in the design of control systems for nuclear reactors (Nieva, 1997). Symmetry considerations allow the decoupling of the linear reactor model into decoupled models of lower order. Thereby, control systems can be developed for each submodel independently.

Similarly, group theoretic principles have been shown to allow the decomposition of solution algorithms of boundary value problems in Nuclear Engineering to be specified over decoupled symmetric domain. This decomposition makes the the problem amenable to implementation for parallel computation (Orechwa & Makai, 1997).

Group theory is applicable in the investigation of the homogenization problem. D. S. Selengut addressed the following problem (Selengut, 1960) in 1960. He formulated the following principle: If the response matrix of a homogeneous material distribution in a volume V can be substituted by the response matrix of a homogeneous material distribution in V, then there exists a homogeneous material with which one may replace V in the core. The validity of this principle is widely used in reactor physics, was investigated applying group theoretic principles (Makai, 1992),(Makai, 2010). It was shown that Selengut’s principle is not exact; it is only a good approximation under specific circumstances. These are that the homogenization recipes preserve only specific reaction rates, but do not provide general equivalence.

Group theory has also been fruitfully applied to in-core signal processing (Makai & Orechwa, 2000). Core surveillance and monitoring are implemented in power reactors to detect any deviation from the nominal design state of the core. This state is defined by a field that is the solution of an equation that describes the physical system. Based on measurements of the field at limited positions the following issues can be addressed:

1. Determine whether the operating state is consistent with the design state.

2. Find out-of-calibration measurements.

3. Give an estimate of the values at non-metered locations.

4. Detect loss-of-margin as early as possible.

5. Obtain information as to the cause of a departure from the design state.

The solution to these problems requires a complex approach that incorporates numerical calculations incorporating group theoretic considerations and statistical analysis.

The benefits of group theory are not restricted to numerical problems. In 1985 Toshikazu Sunada (Sunada, 1985) made the following observation: If the operator of the equation over a volume V commutes with a symmetry group G, and the Green’s function for the volume V is known and volume V can be tiled with copies tile t (subvolumes of V), then the Green’s function of t can be obtained by a summation over the elements of the symmetry group G. Thus by means of group theory, one can separate the solution of a boundary value problem into a geometry dependent part, and a problem dependent part. The former one carries information on the structure of the volume in which the boundary value problem is studied, the latter on the physical processes taking place in the volume. That separation allows for extending the usage of the Green’s function technique, as it is possible to derive Green’s functions for a number of finite geometrical objects (square, rectangle, and regular triangle) as well as to relate Green’s functions of finite objects, such as a disk, or disk sector, a regular hexagon and a trapezoid, etc. Such relations are needed in problems in heat conduction, diffusion, etc. as well.

An extensive discussion of the mathematics and application of group theory to engineering problems in general and nuclear engineering in particular is presented in (Makai, 2011).

The compatibility test was carried out by using the test apparatus with a closed loop shown in Fig. 4. After the specimens were set in the immersion container, aqueous K2MoO4 solution was injected in the closed loop, and the solution was circulated at a constant flow rate. The flow rate was set at about 120 cm3/min, considering the flow velocity assumed in an actual 99Mo production system. The concentration of the solution was adjusted to about 90% of the saturation for the prevention of crystallization, and the temperature of the solution was maintained at about 80°C for the prevention of boiling. As the specimens immersed in the solution, stainless steel SUS304 was used based on the results of the previous immersion tests (Inaba et al., 2009). SUS304 has been used as the structural material of capsules and pipes in JMTR. The size of the specimens was 10Wx30L*1.5T mm. Table 3 shows the chemical composition of a SUS304 specimen. The total immersion time of the specimens was 112.7 days, and the immersion time under flow was 84.5 days out of a total of 112.7 days. The total immersion time was longer than the immersion time under flow because the feed pump was temporarily stopped by the planned blackouts and the pump troubles.

During the test, at regular intervals, the specimen 1 was taken from the immersion container, and the specimen’s weight was measured after washing by pure water and drying, and its surface state was observed. In addition, the aqueous solution was sampled from the closed loop by using one of the syringes, and the pH and molybdenum concentration of the solution were measured, and the solution state was observed.

After the test, the specimen 1 and 2 were taken from the immersion container, and the specimens’ weight was measured, and their surface states were observed. In addition, the aqueous solution was sampled from the closed loop, and the pH and molybdenum concentration of the solution were measured, and the solution state was observed.

C	Si	Mn	P	S	Ni	Cr	Fe
0.06	0.51	0.73	0.026	0.002	8.03	18.07	Balance

(Unit: wt%)

Table 3. Chemical composition of SUS304 specimen

1.4 Results and discussions

The average temperature and flow rate of the aqueous K2MoO4 solution used in the test were 81°C for a total immersion time of 112.7 days and 123 cm3/min for a total immersion time under flow of 85.5 days respectively.

The following sequence of equations can be used in order to calculate the outer surface temperature of the sheath along the heated length of the fuel channel.

Assumption to start the iteration: Tsheaih wall 0 = Tbulk + 50° C

The developed MATLAB code uses an iterative technique to determine the sheath-wall temperature. Initially, the sheath-wall temperature is unknown. Therefore, an initial guess is needed for the sheath-wall temperature (i. e., 50°C above the bulk-fluid temperature). Then, the code calculates the HTC using Eq. (17), which requires the thermophysical properties of the light-water coolant at bulk-fluid and sheath-wall temperatures. Next, the code calculates a "new" sheath-wall temperature using the Newton’s law of cooling shown as Eq. (19). In the next iteration, the code uses an average temperature between the two consecutive temperatures. The iterations continue until the difference between the two consecutive temperatures is less than 0.1 K. It should be noted that the initial guessed sheath-wall temperature could have any value, because regardless of the value the temperature converges. The only difference caused by different guessed sheath-wall temperatures is in the number of iterations and required time to complete the execution of the code.

As mentioned previously, the thermophysical properties of the coolant undergo significant changes as the temperature passes through the pseudocritical point. Since the operating pressure of the coolant is 25 MPa, the pseudocritical point is reached at 384.9°C. As shown in Fig. 16, the changes in the thermophysical properties of the coolant were captured by the Nusselt number correlation, Eq. (16). The Prandtl number in Eq. (16) is responsible for taking into account the thermophysical properties of the coolant. Figure 16 shows the thermophysical properties of the light-water coolant along the length of the fuel channel. The use of these thermophysical properties in the Nusselt number correlation indicates that the correlation takes into account the effect of the pseudocritical point on the HTC between the sheath and the coolant.

1.1.1 Inner-sheath temperature

The inner surface temperature of the sheath can be calculated using Eq. (20). In Eq. (20), k is the thermal conductivity of the sheath, which is calculated based on the average temperature of the outer and inner wall surface temperatures. This inner-sheath temperature calculation is conducted through the use of an iteration, which requires an initial guess for the inner surface temperature of the sheath.

T — T

isheath, wall i isheaih/wall o

ln(r0/ri)

2nLk

In CANDU reactors, three instrumentation systems are provided to measure reactor thermal neutron flux over the full power range of the reactor (Knoll, 2000). Start-up instrumentation covers the eight-decade range from 10-14 to 10-6 of full power; the ion chamber system extends from 10-7 to 1.5 of full power, and the in-core flux detector system provides accurate spatial measurement in the uppermost decade of power (10% to 120% of full power). The fuel channel temperature monitoring system is provided for channel flow verification and for power mapping validation. The self-powered in-core flux detectors are installed in flux detector assemblies to measure local flux in the regions associated with the liquid zone controllers. The flux mapping system uses vanadium detectors distributed throughout the core to provide point measurements of the flux. The fast, approximate estimate of reactor power is obtained by either taking the median ion chamber signal (at powers below 5% of full power) or the average of the in-core inconel flux detectors (above 15% of full power) or a mixture of both (5% to 15% of full power).

2. Several advanced power measuring and monitoring systems

The power range channels of nuclear reactors are linear, which cover only one decade, so they do not show any response during the startup and intermediate range of the reactor operation. So, there is no prior indication of the channels during startup and intermediate operating ranges in case of failure of the detectors or any other electronic fault in the channel. Some new reliable instrument channels for power measurement will be studied in this section.

The purely geometric symmetries of a suitable equation lead to a decomposition (2.16) of an arbitrary function in a function space, and thus the decomposition of the function space itself. The decomposed elements are linearly independent and can be arranged to form an orthonormal system. This can be exploited in the calculations.

In a homogeneous material one can readily construct trial functions that fulfill the diffusion equation at each point of V. For example consider

V2H(r) + Af(r)= 0 (4.12)

£f — Ls + £f D 1. The general solution to (4.12) takes the form of

E tk eiXkr e Wk(e)de k=l lel=1
Hr)	(4.13)

where the weight functions Wk (e) eigenvectors of matrix A:

are arbitrary suitable functions, i2

Afk = Л-k fk.

(4.16)

where Og maps the interval 0 < в < 2n/G into the interval Ig. In this manner we get the irreducible components of the solution as a linear combination of |G| exponential function, it is only the coefficients in the linear combination that determine the irreducible components. The weight function Wk(6) makes it possible to match the entering currents at given points of the boundary. Let в = 0 correspond to the middle of a side. Then choosing

Wk ((e))(6) = Wk 5(6), (4.17)

we get by (4.15) the solution at face midpoints. The last step is the formation of the irreducible components. Observe that in projection (2.20) the solutions at different images of r are used in a linear combination, the coefficients of the linear combinations are the rows of the character table. But in the images (4.16), only the weight function changes. In each Ig interval the image of Wk (e) is involved, which is a Dirac-delta function, only the place of the singularity changes as the group elements map the place of the singularity. A symmetry of the square maps a face center into another face center thus there will be four distinct positions and the space dependent part of the irreducible component of fa will contain four exponentials:

± eiXkX, ±e—iXkX, ±eiXky, ±e—iXky. (4.18)

From these expressions the following irreducible combinations can be formed:

A : cos XkX + cos XkV’; A2 : cos XkX — cos Xky; £1 : sin AkX; £2 : sin Xyy. (4.19)

It is not surprising that when we represent a side by its midpoint the odd functions along the side are missing.

The above method may serve as a starting point for developing efficient numerical methods. The only approximation is in the continuity of the partial currents at the boundary of adjacent homogeneous nodes.

If elements of the function space are defined for all r Є V, and if f, /2 Є Lу, then the following inner product is applicable:

(/1, /2) = Jvft( r)/2(r)d3r. (4.20)

Let /a (r), £ = 1,…, na be a regular representation of group G. Then

furthermore, for the reactions rates formed with the help of the cross-sections in (4.1), similar orthogonality relation holds. For the volume integrated reaction rates we have

Ш const) = 6a,1 $l,1, (4.22)

in other words: solely the most symmetric, one dimensional representation contributes to the volume integrated reaction rates. Note that as a result of the decomposition of the solution or its approximation into irreducible components not only that irreducible components of a given physical quantity (like flux, reaction rate, net current) but also the given irreducible component of every physical quantity fall into the same linearly independent irreducible subspace. As a consequence, the operators (matrices) mapping the flux into net currents (or vice versa) fall into the same irreducible subspace, therefore the mapping matrix automatically becomes diagonal.

Example 4.3 (Symmetry components of boundary fluxes). Consider the flux given along the boundary of a square. The flux is given by four functions corresponding to the flux along the four sides of the square. The flux along a given face is the sum of an even and an odd function with respect to the reflection through the midpoint of the face. The decomposition (2.21) gives the eight irreducible components shown in Figure 2. Note that the irreducible

-1.0

Fig. 2. Irreducible components on the boundary of a square

subspaces а,-, г < 5 are one-dimensional whereas the subspace a = 5 is two-dimensional, and in a two-dimensional representation there are two pairs of basis functions that are identical as to symmetry properties. Thus, we have altogether eight linearly independent basis functions.

The physical meaning of the irreducible components is that the flux distribution of a node is a combination of the flux distributions established by eight boundary condition types. The
component A represents a complete symmetry that is the same even distribution along each side. Component A2 is also symmetric, but the boundary condition is an odd function on each side. Components Bj and B2 represent entering neutrons along one axes and exiting neutrons along the perpendicular axes, a realization of a second derivative with even functions over a face. B2 is the same but with odd functions along a face. Ej and £4 represent streaming in the x and y directions with even distributions along a face, whereas E2 and E3 with odd distributions along a face.

The symmetry transformations of the square, map the functions given along the half faces into each other but they do not say anything about the function shape along a half face. Therefore, the functions in Fig. 2 serve only as patterns, the function shape is arbitrary along a half face. The corresponding mathematical term is the direct product; each function may be multiplied byafunction f (£), — h/2 < £ < +h/2. It is well known that a function along an interval can be approximated by a suitable polynomial (Weierstrass’s theorem). We know from practice that in reactor calculations a second order polynomial suffices on a face for the precision needed in a power plant.

The invariant subspace means that the boundary flux, the net current, the partial currents must follow one of the patterns shown in Figure 2, the only difference may be in the shape function f (£), — h/2 < £ < +h/2. This means the a constant flux may create a quadratic position dependent current, but the global structure of the flux, and current should belong to the same pattern of Figure 2.

Moreover, if we are interested in the solution inside the square, its pattern must also be the same although there the freedom allows a continuous function along 1/8-th of the square. These features are exploited in the calculation. □

In which two states the positive reactivity overcomes the temperature coefficient of reactivity (aT):

Increasing the power and temperature of reactor core might decrease concentration of boric acid. Accordingly this event might cause to inject positive reactivity.

In addition for the reactors which apply fuels including Pu, because of having a resonance for Pu in thermal neutrons range so through increasing core’s temperature the related resonance is broadened and absorbs more neutrons and because of Pu is fissionable therefore fissionable absorption occurs and is caused excess reactivity.

Also either accident or unfavorable issues as a feedback can be considered. Accidents of a nuclear reactor are totally classified based on following:

1. Over power accident.

2. Under cooling accident.

Each mentioned issues are divided to other sub issues. Over power accident is due factors such as:

1. Control rod withdrawal including uncontrolled rod withdrawal at sub critical power and uncontrolled rod withdrawal at power that will cause power excursion.

2. Control rod ejection.

3. Spent fuel handling.

4. Stem line break.

5. External events such as earthquake, enemy attack and etc.

In each mentioned issues the positive reactivity to nuclear reactor core can be injected. But there is another important accident that is: under cooling accident. Under cooling accident is classified to three sub accident among: LOCA (loss of coolant accident), LOHA (loss of heat sink accident) and LOFA (loss of flow accident).

LOCA accident from loosing coolant is derived. This event in PWR reactor can occur through breaking in primary loop of reactor either hot leg or cold leg.

In this state the existent water in the primary loop along with steam are strongly leaked that blow down event occurs. But when the lost water of primary loop through RHRS (Residual and Heat Removal System) including HPIS (High Pressure Injection System), LPIS (Low Pressure Injection System) and Accumulator (passive system) are filled this process is entitled Refill. When the primary is filled and all the lost water in it is compensated then the reactor sets in the normal status.

In the blow down status the raised steam is due loss of pressure in primary loop and moving the situation of reactor’s primary loop from single phase to two phase flow.

LOHA accident from loosing heat sink in reactor is derived. Heat sink is as steam generator in nuclear reactor. This event when occurs heat exchanging between primary and secondary loops are not done.

This event might occur through lacking water circulation in the secondary loop of reactor. Not circulating the water might occur through closing either block valve (which sets after demineralizer tank) or other existent valves in the secondary loop.

LOFA accident from disabling and loosing the pumps in either primary loop or secondary is derived. In case either primary loop’s pump or secondary encounter with problems LOFA accident might occur.

In the secondary loop the main feed water pump has duty of circulation of feed water to steam generator and sets after condenser pump.

In view of dynamically stability of a nuclear reactor, there is a stable system so that an excess reactivity is injected to it and it able to be stabled again in shortest time. The stability of linear systems in the field of complex numbers by defining the polarity of closed loop transfer’s function is determined.

In case all the polarities are in the left side of imagine page then system will be stabled. In the time field the stability definition means system’s response to each input will be definitive. In the matrix form all the Eigen values of system have real negative part.

Power monitoring using thermal power produced by reactor core is a method that is used in many reactors. To explain how the method is used for reactor power measurement, a research reactor is studied in this section. In IPR-R1, a TRIGA Mark I Research Reactor, the power is measured by four nuclear channels. The departure channel consists of a fission counter with a pulse amplifier that a logarithmic count rate circuit. The logarithmic channel consists of a compensated ion chamber, whose signal is the input to a logarithmic amplifier, which gives a logarithmic power indication from less than 0.1 W to full power. The linear channel consists of a compensated ion chamber, whose signal is the input to a sensitive amplifier and recorder with a range switch, which gives accurate power information from source level to full power on a linear recorder. The percent channel consists of an uncompensated ion chamber, whose signal is the input to a power level monitor circuit and meter, which is calibrated in percentage of full power. The ionization chamber neutron detector measures the flux of neutrons thermalized in the vicinity of the detector. In the present research, three new processes for reactor power measurement by thermal ways were developed as a result of the experiments. One method uses the temperature difference between an instrumented fuel element and the pool water below the reactor core. The other two methods consist in the steady-state energy balance of the primary and secondary reactor cooling loops. A stainless steel-clad fuel element is instrumented with three thermocouples along its centerline in order to evaluate the reactor thermal hydraulic performance. These processes make it possible on-line or off-line evaluation of the reactor power and the analysis of its behavior.

In the new solution irradiation method, a solution target including natural molybdenum such as an aqueous solution of a molybdenum compound (aqueous molybdenum solution) is irradiated with neutrons in a testing reactor, and 99Mo is produced by the 98Mo (n, y) 99Mo reaction. This new method is the improved type of the solid irradiation method, and it is possible to enhance the 99Mo production compared with the solid irradiation method. The solution irradiation method has the following advantages compared with the solid irradiation method:

1. It is easy to increase the irradiated volume by using a capsule with larger volume than that of a rabbit. The rabbit is a small sized (150 mm length) capsule (Inaba et al., 2011).

2. The separation and dissolution processes after the irradiation are not necessary because the irradiation target is an aqueous solution.

3. The amount of generated radioactive waste is smaller than that of the solid irradiation method.

	99Mo production methods
Items	Fission method ((n, f) method)	Neutron capture method ((n, у) method)
		Solid irradiation method	Solution irradiation method
<Irradiation target>	Enriched 235U	Natural Mo	Natural Mo
•Chemical type	•U-Al alloy, UO2	• MoO3, metal Mo	•Molybdate
•Form	•Foil, pellet	•Powder, pellet, metal	•Aqueous solution
•Quality control	•Complex	•Complex	•Simple
<Irradiation container>	Rabbit (30 cm3)	Rabbit (30 cm3)	Capsule (1,663 cm3)
<Irradiation>
•Pre-process of irradiation	•Adjustment of target and enclosing with container	•Adjustment of target and enclosing with container	•Adjustment of target
•Reaction	•235U (n, f)	•98Mo (n, у) 99Mo	•98Mo (n, у) 99Mo
of 99Mo production
•Irradiation time	• 5-7 days	•5-7 days	•5-7 days
•Collection of target	•Batch collection	•Batch collection	• Continuous or batch collection
•Post-process of irradiation	•Isolation in hot lab. (Complex)	•Dissolution in hot lab. (Relatively simple)	•No special treatment
Generated 99Mo
•Specific activity	•370 TBq/g-Mo	•37-74 GBq/g-Mo	•37-74 GBq/g-Mo
•Activation by-product	•Quite many (131I, 103Ru, 89Sr, 90Sr, etc.)	•92mNb	•92mNb, 14C, 42K, etc. (depending on a target solution)
<Mo adsorbent>	Alumina (2 mg-Mo/g-AhO3)	PZC (250 mg-Mo/g-PZC)	PZC (250 mg-Mo/g-PZC)
<Radioactive waste>	Rabbits with FPs and Pu (Generation every one irradiation)	Rabbits (Generation every one irradiation)	Capsule (Lifetime: about 15 operation cycles)
<Production in Japan>	Difficult	Possible	Possible

Table 1. Comparison between three 99Mo production methods

In this new method, efficient and low-cost 99Mo production compared with the conventional 99Mo production can be realized by using the (n, y) reaction and PZC. This new method aims to provide 100% of the 99Mo imported into Japan.