638 MODEL OF CLADDING FAILURE ESTIMATION UNDER

MODEL OF CLADDING FAILURE ESTIMATION UNDER MULTIPLE CYCLIC REACTOR POWER CHANGES S. N. Pelykh, M. V. Maksimov, V.E. Baskakov Odessa National Polytechnical University, Odessa, Ukraine Nuclear reactor operating modes under multiple cyclic power changes have been promoted recently, and fuel element cladding behavior under the multiple cyclic power changes has been widely known as a key issue in terms of rod design and reliability. A model of nuclear reactor fuel rod cladding failure estimation under multiple cyclic power changes is proposed. The model is built on the basis of the following admissions of the energy version of creep theory: processes of cladding creep and destruction proceed together and affect each other; intensity of creep process is estimated by specific dispersion power W (τ), while intensity of destruction – by specific dispersion energy A (τ) accumulated during time τ. Having calculated the equivalent stress and the rate of equivalent creep strain, the condition of fuel rod cladding failure used on the basis of the energy version of the theory of creep gives us a criterion to decide if a multiple cyclic power change operating mode is permissible for a given variant of power history and coolant conditions. Operation of nuclear power units of Ukraine in the variable part of electric loading schedule (variable loading mode) has become actual recently, that means there are repeated cyclic nuclear reactor (NR) capacity changes during NR normal operation [1]. As is known, utilization factor of maximum capacity UFMC is obtained as n

UFMC =

∑( ∆τ i =1

i

⋅ Ni )

T ⋅N

,

(1)

where ∆τ i – NR operating time at the capacity of Ni ; T – total NR operating time; N – maximum NR capacity (100 %). For instance, let's consider the following NR loading modes: 1) Stationary operation at 100 % NR capacity level, UFMC = 1. 2) The NR works at 100 % capacity level within 5 days, then the reactor is transferred to 50 % capacity level within 1 hour. Further the NR works at the capacity level of 50 % within 46 hours, then comes back to 100 % capacity level within 1 hour. Such NR operating mode will be designated as the (5 d. – 100 %, 46 h. – 50 %) weekly load cycle, UFMC = 0.860 (Fig. 1, line 1). 3) The NR works at 100 % capacity level within 16 hours, then the reactor is transferred to 75 % capacity Fig. 1. 1 – (5 d. – 100 %, 46 h. – 50 %) weekly load cycle; level within 1 hour. Further the NR works at 75 % 2 – (16 h. – 100 %, 6 h. – 75 %) daily load cycle. capacity level within 6 hours, then comes back to 100 % capacity level within 1 hour. Such NR operating mode will be designated as the (16 h. – 100 %, 6 h. – 75 %) daily load cycle, UFMC = 0.927 (see Fig. 1, line 2). 4) The NR works according to the (16 h. – 100 %, 6 h. – 75 %) daily cycle in week-days, but the NR capacity decreases to 50 % level within last hour of every fifth day of a week. Further the reactor works during 47 hours at 50 % capacity level and, at last, within last hour of every seventh day the NR capacity rises to the level of 100 %. Such NR operating mode will be designated as the combined (5 d. – 100 % + 75 %, 2 d. – 50 %) weekly load cycle, UFMC = 0.805. Fuel element (FE) operation is characterized by long influence of high-level temperature-power stressing leading to uncontrollable cladding material creep processes causing, after a while, its destruction, and fission products enter the circuit in the quantities exceeding both operational limits and limits of safe operation. In this connection, estimation of cladding integrity time for a NR variable loading mode, taking into account some appointed criteria, becomes one of key problems of FE designing and active core operational reliability analysis. Difficulty of this problem is caused by the fact that cladding material creep modeling under the conditions corresponding to operational variable load modes is inconvenient or impossible as such tests can last for years. Besides, as all power history affects the cladding, it is incorrect to transfer experimental stationary and emergency operation cladding material creep data onto the FE cladding working at variable loading. Emergency NR operation leading to cladding material plastic deformation is not studied here, therefore the hot plasticity (stress softening) arising at the expense of yield stress decrease under emergency cladding temperature rise, is not considered.

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So, to estimate FE cladding running time under multiple cyclic NR power changes, it is enough to calculate the energy accumulated during the creep process, by the moment of cladding failure and spent for cladding material destruction [2]. According to Energy variant of the theory of creep [3], the energy spent for FE cladding material destruction is called as Specific Dispersion Energy (SDE) A (τ). The offered method to analyze FE cladding running time at variable loading is based on the following assumptions of the energy variant of creep theory: creep and destruction processes proceed in common and influence against each other; at any moment τ creep process intensity is estimated by Specific Dispersion Power (SDP) W (τ), while intensity of failure is estimated by the SDE A (τ) accumulated during the creep process by the moment τ: τ

A(τ ) = ∫W (τ ) ⋅ dτ .

(2)

0

Let’s enter the cladding material Failure Parameter ω (τ ) into the analysis:

ω (τ ) =

A(τ ) , A0

(3)

where A0 – the SDE at the moment of cladding material failure beginning, known for the given material either from experiment, or from calculation, J/m3; ω = 0 – for the intact material, ω = 1 – for the damaged material. The SDP W(τ) standing in (2) is defined by the following equation [2]: W (τ ) = σ e ⋅ p e ,

(4)

where σe – equivalent stress, Pa; p e – rate of equivalent creep strain, с-1. The condition of cladding material failure is derived from (2) – (4): τ

ω (τ ) = ∫ 0

σ e ⋅ p e A0

⋅ dτ = 1

(5)

The equivalent stress σe and the rate of equivalent creep strain p e are calculated by the light water fuel analysis code FEMAXI [4]. The code FEMAXI analyzes changes in the thermal, mechanical and chemical state of a single fuel rod and interaction of its components in a given NR power history and coolant conditions. The code analytical scope covers normal operation conditions and transient conditions such as load-following and rapid power increase in a highburnup region of over 40 … 50 MWd/kg-U. The fuel temperature calculation was carried out with the difference between the numerical solution and analytical solution not exceeding 0.1 %. The numerical error arising in the form of residue from iterative creep calculation on each time step was not estimated as in most cases this error is exceeded by other uncertainties, first of all by thermal conductivity model error [4].

Fig. 2. Dependence of the SDE A (N) from the number of daily cycles N. 1 – stationary NR operation at 100 % power level; 2 – daily cycle: (16 h. – 100 %; 6 h. – 75 %); 3 – daily cycle: (16 h. – 100 %; 6 h. – 50 %); 4 – daily cycle: (16 h. – 100 %; 6 h. – 25 %).

Fig. 2 shows the dependence of the SDE A (N) for zircaloy (stress relieved zircaloy) cladding from the number of daily load cycles N, for the daily load cycle: (16 h. – 100 %; 6 h. – k·100 %), k = 1; 0.75; 0.5; 0.25. NR type: PWR. As a rule, the creep phenomenon has three characteristic stages: unsteady, steady and rapid creep (the last one – predestruction stage) [2]. The borders of the characteristic creep stages for lines (1) – (4) are shown in Table 1. The number of daily cycles Ne,0 that the cladding can withstand prior to the beginning of the rapid creep stage, expressed in effective days, is defined from the following equation: Ne,0 = N0 ⋅ UFMC,

(6)

where N0 – the number of calendar daily cycles prior to the beginning of the rapid creep stage; UFMC – defined from (1). 639

Table 1. Characteristic creep stages Line 1 (100 %) 2 (75 %) 3 (50 %) 4 (25 %)

Сreep Stage Steady Creep A1(100)…A1(702) A2(100)…A2(760) A3(100)…A3(820) A4(100)…A4(860)

Unsteady Creep A1(0)…A1(100) A2(0)…A2(100) A3(0)…A3(100) A4(0)…A4(100)

Rapid Creep A1(702)…A1(1200) A2(760)…A2(1200) A3(820)…A3(1200) A4(860)…A4(1000)

The equivalent creep strain pe for zircaloy cladding, for all daily load modes shown in Fig. 2, gradually increases and the hysteresis decrease of pe can be seen at the last stage of creep. Then, after the hysteresis decrease, pe starts to grow fast and achieves inadmissibly great values from cladding reliability point of view (Table 2). The UFMC, the SDE accumulated before the rapid creep stage starts A0, the number of calendar daily cycles N0 and the number of effective daily cycles Ne,0 for lines 1 – 4 are shown in Table 2 also. Cladding material: stress relieved zircaloy, NR type: VVER. Table 2. Characteristic parameters for Lines (1) - (4) Line 1 (100 %) 2 (75 %) 3 (50 %) 4 (25 %)

pe , % (after 1200 daily cycles) 6.75 4.56 3.25 2.84

UFMC

A0, MJ/m3

N0, days

1 0.93 0.85 0.78

0.522 0.559 0.639 0.585

702 760 820 860

Ne,0, eff. days 702 704 700 672

The following qualitative conclusions can be drawn on the basis of the data shown in Table 2: cladding running time for (16 h. – 100 %; 6 h. – 75 %) cycle elongates a little, while cladding running time for (16 h. – 100 %; 6 h. – 25 %) cycle considerably decreases in comparison with the stationary NR operation at 100 % power level. If the number of history points increases from 4 to 8 a day, the estimated cladding running time changes not more than on 0.5 % for the daily cycle case. Hence, the appointed number of history points for the daily cycle case allows us to calculate A (N) with sufficient accuracy. Similarly, set of 56 history points a week for (5 d. – 100 %, 46 h. – k·100 %) weekly cycle allows us to calculate A (N) with sufficient accuracy (k = 1; 0.75; 0.5; 0.25). Like the above described analysis for the case of daily cycle, it was found, that cladding running time, expressed in effective days, for (5 d. – 100 %, 46 h. – 50 %) weekly cycle elongates a little, while cladding running time for (5 d. – 100 %, 46 h. – 25 %) weekly cycle considerably decreases in comparison with the stationary NR operation at 100 % power level. At last, Fig. 3 shows the dependence of the SDE A (N) from the number of weekly load cycles N for the cases: stationary NR operation at 100 % power level; Fig. 3. Dependence of the SDE A (N) from the number of (16 h. – 100 %; 6 h. – 75 %) daily cycle; (5 d. – 100 %, weekly cycles N. 1 – stationary NR operation at 100 % power level; 2 – (16 h. – 100 %; 6 h. – 75 %) cycle; 3 – 46 h. – 50 %) weekly cycle; (5 d. – 100 % + 75 %; 2 d. – 50 %) combined cycle. Cladding material: stress relieved (5 d. – 100 %, 46 h. – 50 %) cycle; 4 – (5 d. – 100 % + zircaloy, NR type: VVER. + 75 %; 2 d. – 50 %) combined cycle. Using the equation (6) and the data shown in Fig. 3, it is possible to conclude, that cladding running time for (5 d. – 100 % + 75 %; 2 d. – 50 %) combined cycle is maximal and equals to 102.3 effective weekly cycles, while cladding running time for the stationary NR operation at 100 % power level is minimal and equals to 100.0 effective weekly cycles (Table 3). A concrete nuclear reactor’s FE cladding running time to be estimated, it is necessary to assign the constructional and regime characteristics of this NR. Then, using the offered technique, the values of cladding material failure parameter ω (τ ) can be plotted in the NR operating charts for typical power histories.

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Table 3. Cladding running time as a function of NR loading mode

Parameter UFMC N, weeks Nе, eff. weeks

Stationary NR operation at 100 % power level 1 100.0 100.0

NR loading mode (5 d. – 100 %, (16 h. – 100 %; 46 h. – 50 %) 6 h. – 75 %) weekly cycle daily cycle 0.860 0.927 117.0 109.9 100.6 101.9

(5 d. – 100 % + 75 %; 2 d. – 50 %) combined cycle 0.805 127.1 102.3

The proposed method to analyze cladding running time at variable loading can appear irreplaceable as the necessary conclusion can hardly be obtained by the exact analytical decision of the creep task or by experiments. Acknowledgments The authors are deeply indebted to Dr. Motoe Suzuki, Japan Atomic Energy Agency for his consultations about some aspects of use of the FEMAXI code. REFERENCES 1. Мaksimov М., Maslov O., Fridman N. Determination of the operation efficiency criterion for a nuclear power plant with the VVERs working in the variable part of the electric loading schedule // Works of Odessa Polytech. Univ. 2001. - Vol. 2 (14). - P. 78 - 80 (in Russian). 2. Nemirovsky Y. About an estimation of construction safe operation time // Proc. of the Int. Workshop RDAMM-2001. - Novosibirsk, 2001. - P. 328 - 333 (in Russian). 3. Sosnin O., Gorev B.V., Nikitenko A. Energy variant of the theory of creep. - Novosibirsk: The Siberian branch of the Russian Academy of Sciences, The Thermodynamics Institute, 1986. - 95 p.(in Russian). 4. Motoe Suzuki. Light Water Reactor Fuel Analysis Code FEMAXI-V (Ver. 1). - Tokai: Japan Atomic Energy Research Institute, 2000. - 285 p.

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