Core designing of a new type of TVS-2M FAs: neutronics and thermal-hydraulics design basis limits

Abstract

One of the most important aims of this study is to improve the core of the current VVER reactors to achieve more burn-up (or more cycle length) and more intrinsic safety. It is an independent study on the Russian new proposed FAs, called TVS-2M, which would be applied for the future advanced VVERs. Some important aspects of neutronics as well as thermal hydraulics investigations (and analysis) of the new type of Fas are conducted, and results are compared with the standards PWR CDBL. The TVS-2M FA contains gadolinium-oxide which is mixed with UO₂ (for different Gd densities and U-235 enrichments which are given herein), but the core does not contain BARs. The new type TVS-2M Fas are modeled by the SARCS software package to find the PMAXS format for three states of CZP and HZP as well as HFP, and then the whole core is simulated by the PARCS code to investigate transient conditions. In addition, the WIMS-D5 code is suggested for steady core modeling including TVS-2M FAs and/or TVS FAs. Many neutronics aspects such as the first cycle length (first cycle burn up in terms of MW_thd/kgU), the critical concentration of boric acid at the BOC as well as the cycle length, the axial, and radial power peaking factors, differential and integral worthy of the most reactive CPS-CRs, reactivity coefficients of the fuel, moderator, boric acid, and the under-moderation estimation of the core are conducted and benchmarked with the PWR CDBL. Specifically, the burn-up calculations indicate that the 45.6 d increase of the first cycle length (which corresponds to 1.18 MW_thd/kgU increase of burn-up) is the best improving aim of the new FA type called TVS-2M. Moreover, thermal-hydraulics core design criteria such as MDNBR (based on W3 correlation) and the maximum of fuel and clad temperatures (radially and axially), are investigated, and discussed based on the CDBL.

Keywords

TVS-2M FAs / core design basis limits / VVER-1000 / analysis / mixture of uranium-gadolinium oxides fuels / thermal-hydraulics / PARCS / WIMS-D5

Introduction

The cycle-length of the current Russian PWR reactors (e.g., VVER-1000) is about 293 d (7032EFPH), but a new FA, called TVS-2M (which contains mixed uranium-gadolinium oxide with different Gd densities and U enrichments), is proposed to reduce the electric energy production costs, increase fuel cycle lengths, and fuel burn ups [¹–⁴] with more intrinsic safety. Without doubt, every plant must work at a minimal cost to increase the NPP load factor, to achieve the goal of increasing fuel campaign duration, more fuel burn-up as close to safety limits as possible. Use of mixed uranium-gadolinium oxide makes neutron radial flattening which causes uniform distribution of neutron flux, and at the same time, causes a slower fission rate for achieving greater fuel cycle lengths. Also, this newly proposed FA results in larger negative reactivity coefficients at the BOC which creates more self-regulation without increasing the number of control rods and boric acid [³]. The efficiency and quality of the operation and safety of the core are affected by the manner and methods of compensating the excess reactivity.

Today, gadolinium oxide is widely used as a burnable absorber due to its unique chemical and nuclear properties in thermal and epithermal regions which are superior to the other chemical elements. Gadolinium has seven isotopes, as listed in Table 1.

But there are two important gadolinium isotopes, Gd-155 and Gd-157, which contain a high absorption cross section in the thermal region [⁵]. Besides, Gd₂O₃ mixed with UO₂ may be used as a burnable absorber in the fuel rod. Use of uranium-gadolinium decreases the possibility of the uranium consumption, especially in the thermal and resonance regions which cause slower thermal fission rate; and hence more cycle-length. But, its abnormality in the core safety margins and criticality conditions must be studied. Large amounts of boron in the moderator (as boric acid) decrease the negative temperature coefficient of reactivity [⁶], especially at the BOC. Using the mixed gadolinium fuel rods leads to a reduction in boron concentration which decreases operation cost.

The first inquiry of the new proposed fuel type is about maintaining under-moderated core at the startup and during the cycle which is under investigation in this paper. The second goal would be negative temperature coefficients as well as core criticality. The third aim would be the PWR criteria margins. In other words, the new core should maintain safety margins which are given in Table 3.11 of Ref. [⁶].

The new generation FAs, called TVS-2M, will replace the current BNPP VVER-1000 FAs, which is called “TVS core” here, by the Russian owners. Therefore, in this paper, independent investigations have been made on the new type of FAs which will be used in the next generation VVERs reactors. In average, new TVS-2M FAs have higher enrichment, but in comparison with the current TVS FAs, their heights are 15 cm increased.

In the upcoming sections, the new core configuration (filled with TVS-2M FAs) as well as the CPS-CR patterns will be illustrated. Then, under-moderation study as well as criticality approach would be investigated using WIMS and CITATION codes. Besides, the main criteria of the PWR core design limits are studied, and compared with the criteria values mentioned in Table 3.11 of Ref. [⁶]. It should be emphasized here that the TVS-2M FAs are modeled by the SARCS package [⁷], and the PMAXS format for the three states of CZP, HZP as well as HFP are generated, and the whole core is simulated by the PARCS code to investigate transient conditions [⁷]. Finally, a description on the MDNBR and a useful formula are given. Briefly, many neutronics and thermal-hydraulics effects are studied herein.

Core configurations including TVS and TVS-2M FAs

Details of the core configuration of the TVS-2M and the current BNPP VVER-1000/443 including TVS FAs are exhibited in Fig. 1, in which Fig. 1(a) shows the various FA enrichments with and without BARs (with various burnable absorber’s density) [⁸].

In addition, Fig. 1(b) contains the TVS-2M core patterns including 5 types of FAs enrichments (and Gd concentrations), but without BARs [⁹]. Moreover, there are 10 CPS-CRs banks whose patterns have changed a little in both TVS and TVS-2M cores.

The lower notations of Fig. 1(b) show the type of FAs including their enrichment and/or gadolinium concentrations whose details are given in Fig. 2. Two types of U13 and U22 contain the average enrichment of 1.3% and 2.2%, respectively and do not contain gadolinium oxide. The other three types of U30Y9, U39A9, and U39B6 contain different gadolinium-uranium oxides with different enrichment and Gd concentration. Their FA average enrichments are 2.98%, 3.89%, and 3.9%, respectively.

Finally, the distribution of CPS-CR for the TVS-2M and TVS cores are plotted in Fig. 3. The upper number in Fig. 3 is the number of FA, and the lower number is the bank of the CPS-CR number. There are 10 different CPS-CR banks in both cores in which a minor difference exists for CPS-CR patterns between the TVS and TVS-2M cores. It is emphasized here that the new core does not contain BARs, instead, the gadolinium oxide is mixed with the uranium oxide to obtain neutron flattening and/or a slower thermal fissioning, to achieve more burnups and more fuel cycle length. At the moment, as mentioned before, for simplicity, the current core which is filled by the TVS FAs is called the “TVS core,” and the newly proposed core which will be filled by the TVS-2M FAs called the “TVS-2M core.” For comparison, the main parameters of the TVS-2M and TVS FAs including their sizes, fuel meat, clad thickness, and other useful physical parameters are listed in Table 2 and Fig. 4.

Thermal-hydraulics core design criteria for the new TVS-2M FA

In the current section, the details of calculation for the PWR thermal-hydraulics design criteria will be explained based on Table 3.11 in Ref. [⁶].

Under-moderated core estimation

The first criterion of each core would be the estimation of the “core under-moderation.” This criterion should be obtained to reach self-regulating of each core, and it is an IAEA license for all PWRs. In other words, the core should be designed to remain in the under-moderated region, as an intrinsic safety margin, to make core self-regulation. Hereby, the moderator coefficient of the reactivity a_T = ∂r⁄∂T (where r and T are reactivity and moderator temperature, respectively) should be less than zero as a constraint. As is known, for a worse accident, ∂T increases, therefore ∂r should be less than zero so that a_T remains negative, which is called the under-moderated core. In other words, a negative a_T makes power decrease in the primary circulation time scale, and therefore makes core self-regulating or core intrinsic safety.

As the reactor designer changes the amount of moderator-to-fuel ratio (that is, N_m/N_u changes where N_m and N_u are the moderator and fuel atomic number densities, respectively), neutron leakage, resonance scape probability (p), and the fuel utilization factor (f) will change, which causes the multiplication factor (k_eff) to change.

Full studies on the annular rods in the BNPP were conducted, and its core modeling, the pin pitch optimization for both solid (UO₂) and annular pins (UO₂) were investigated earlier [¹⁰]. In addition, a few thermal-hydraulics constraints were studied to achieve a possibly proper configuration of the current TVS core [¹¹, ¹²].

In the current core design study, it is known that the core radial, the axial dimensions, and the pitch of the FAs (23.6 cm) do not change. Besides, by focusing on the details in Table 2 and Fig. 4, it is also found that the fuel rod pitch of the TVS-2M and its overall width are the same as TVS pins. Moreover, the overall height of both types of FAs are the same (4570 mm), but the active fissioning of the TVS-2M rods are higher than that of the “TVS-2M core” (c.f., Table 2). In addition, the cladding thicknesses of both pin types are the same (0.65 mm). Moreover, the outside diameters of the fuel rod are the same for both types (9.1 mm). Therefore, the pin pitch-to-diameter (P/D) for both FAs should be the same. Furthermore, the numbers of unheated tubes (which may contain guide tubes for control rods movement and/or contain BARs in the TVS core) are the same. But, there are not any BARs in the TVS-2M core, instead, there are (Gd+ U)O₂ fuels so that the overall weight of UO₂ in the TVS-2M FA is close to 525 kg.

Based on the above conditions (the same P/D, the fixed cores dimensions as well as fixed overall dimensions of FAs, and the same FAs pitch), it is emphasized here that it is sufficient enough to calculate moderator-to-fuel weights for under-moderation estimation. Specifically, the ratio of moderator-to-fuel weight is verified at CZP, HZP, and HFP. Because, keeping a core in the under-moderated region could be challenging specially during start-up condition when all components are going from a cold state (CZP) to a hot state (HZP and HFP). For instance, variations of fuel and moderator densities, boric acid decreasing, temperature increasing, Xe saturation, pressure variations, nitrogen insertion, and fission fragment buildup are major challenges for transferring a core from CZP to HFP.

It is very important to emphasize here that the TVS core under-moderation has been absolutely verified previously both theoretically [¹⁰], and operationally (because this core is now under IAEA operational license, and is tested during major transient conditions such as start-up and shut-down). Hence, the new TVS-2M core under-moderation has been verified based on the current TVS core.

According to Fig. 2 and Table 2 which contain details of each type of FA, it is found that the total UO₂ weight of each TVS-2M FA is 525 kg. Therefore, the total uranium weight of the TVS-2M core is close to 163 × 525 × 238/270= 75433 kg at CZP (where core contains 163 FAs; atomic and molecular weights of U and UO₂ are approximately 238 and 270, respectively). Briefly, variations of the core materials and/or densities due to the three different states of CZP, HZP, and HFP (but still at the BOC) are simulated and supposed as:

CZP: water density ≈ 1g/cm³, free of xenon, maximum boric acid= 8.4 g/kg water, fresh fuel without fission fragment and poisons (e.g., Xe and Sm)

HZP: water density ≈ 0.75 g/cm³, free of xenon, boric acid concentration= 6.4 g/kg water, approximately fresh fuel but still approximately free of poisons such as Xe and Sm.

HFP: water density ≈ 0.67 g/cm³, boric acid concentration ≈ 6.4 g/kg water, approximately fresh fuel (neglected fission fragments at the BOC) but full of Xe and Sm (Xe reaches saturation after about 4 h operation).

As has been mentioned before, only comparison of the weights of fuels are sufficient for moderator-to-fuel ratio estimation. Table 3 contains the results of the TVS-2M and the TVS (under-moderated) cores at three different states (at BOC) when the moderator-to-fuel ratio may rapidly change due to major transient. It is emphasized here that the moderator volumes do not efficiently change for both cores.

The total weight of the uranium is equal to 69568 kg and 75433 kg for the TVS and TVS-2M cores, respectively. Therefore, the “relative” moderator-to-fuel ratio is increased about 8% at CZP (c.f., the last column of Table 3). In addition, moderator density variations in the hot states and/or Xe build-up do not significantly change for both cores, and the “relative” moderator-to-fuel ratio does not significantly change in HZP and HFP for both cores. The major result of Table 3 is that the relative moderator-to-fuel ratio for both cores has only 8% difference at the CZP, and 6% difference at HZP as well as HFP (HFP state is the most important state for an operating reactor).

Figure 9-5 of Ref. [¹³] displays the variation of the multiplication factor versus x for 2% enriched homogeneous core. Explicitly, variations of the important parameters which affect the multiplication factor are given in Ref. [¹³]. The remaining parameters in the six factor formula are also affected by the fuel-to-moderator ratio (x), but to a smaller extent than f, p, and ϵ. By increasing water temperature, the density of water will decrease, and therefore, x will increase. The maximum value of multiplication factor occurs at x = 0.36 and for x greater than 2.5 the multiplication factor would be less than unit (the core would be sub-critical). Therefore, an under-moderated region is within 0.36<x<2.5, and the over-moderated region is within 0.07<x<0.36.

In practice, water-moderated reactors are designed with a fuel-to-moderator ratio so that the reactor operates in an under-moderated region (0.36<x<2.5). If the core were under-moderated, an increase in temperature would increase the x = N_u/N_w (where N_u and N_w are fuel and moderator atomic densities, respectively) due to the water density decreasing. This increase in x = N_u/N_w causes a decrease in k_eff which causes a negative reactivity addition. Power decreasing due to negative reactivity causes a lower temperature which finally causes the reactor to become self-regulating. On the contrary, if the core is in the over-moderated region (0.07<x<0.36), the same increase in coolant temperature results in an addition of positive reactivity, and finally a dangerous situation. Therefore, an under-moderated core condition is the design-license for all operating reactors.

Now, by focusing on Table 3 and on the HFP, it is seen that the moderator/fuel ratio is 1.38 and 1.46 for the TVS and TVS-2M cores, respectively which correspond to x = 0.725 and 0.676.

In other words, based on results of Table 3, there is a maximum of 6% difference in the HFP operation between the TVS and TVS-2M cores. The TVS core with x = 0.725 is shifted at maximum to x = 0.676 for the TVS-2M core due to the mentioned 6% difference of x. As a result, based on Figure 9-5 of Ref. [¹³], the new TVS-2M core is still in the under-moderated region. Therefore, since the current under-operating TVS core has the IAEA license (based on the under-moderated condition), the future TVS-2M core will remain under-moderated, and the 6% difference of x does not majorly affect the under-moderated condition. In addition, the new TVS-2M core is quite controllable by the boric acid insertion (and/or nitrogen insertion), and basically the TVS-2M core will not be over-moderated by this 6% difference regarding the current TVS core.

Self-regulation of the core with negative reactivity feed backs

In Subsection 3.1, it has been estimated that the TVS-2M core has been under-moderated. But, here, it is numerically demonstrated that the TVS-2M core is under-moderated, according to the negative reactivity feedbacks calculations. In other words, by calculating the moderator, fuel, and boric acid reactivity coefficients the self-regulation conditions are verified.

The whole core has been simulated using WIMS and CITATION codes to obtain the exact values of the desired parameters. The details of calculational procedures were given in Ref. [¹²], while the results of the reactivity coefficients versus burnup, for the TVS-2M core, will be given in Subsection 4.6 in this paper.

MDNBR calculations for the TVS-2M core

MDNBR is one of the main criteria for each core basis design limits calculation, and the MDNBR limit for the TVS-2M core has been examined. It is hoped that DNB in a PWR (or dry-out in a BWR) can occur above the active FA region so that the boiling crisis does not happen. Now, suppose a single (hottest) channel which contains fuel rods and its associated coolant. The linear heat rate (q') at which DNB occurs decreases with Z_DNB (it is schematically shown in Fig. 5, and will be explained later). Moreover, as is known, the linear heat rate of fuel rod has a near-cosine shape in the vertical position based on the neutronics behavior (in fact, a minor change from cosine shape occurred in a real heterogeneous core due to different moderator densities at the top and the bottom of a fuel rod which is also exhibited in Fig. 5.).

Obviously, for Z<Z_DNB, there is “single phase heat transfer coolant flow” (or heat transfer in nucleate boiling regime) which is the favorite. But, for Z>Z_DNB coolant flow falls into another regime which is called the “film boiling,” and therefore, a poor heat transfer occurs. As stated earlier, it is hoped that Z_DNB can occur upper than the active fission length of the fuel rod.

One of the most efficient methods for calculating the MDNBR is the W3 correlation which will follow in the current subsection for the TVS-2M core. There are four terms for computing {q^{″}}_{\mathrm{DNB}} (departure from nucleate boiling heat flux), based on the W3 correlation as following:

1) System pressure which is called P-term. The numerical approximation for P-term is in the form of

P-\mathrm{term}=\left(a–bP\right)+\left(a\prime –b\prime P\right)\exp (\left(a″–b″P\right)X_{\mathrm{e}},

where a, b, a', b', a", b" are constants, and X_e is the “thermo-dynamics steam quality” which is found by X_e = [(h(z) – h_f))⁄((h_g – h_f)] so that h_g and h_f are the enthalpy of gas and water, respectively which are evaluated at system pressure P. P and X_e vary along with the axial direction, and hence P-term varies across the axial position. For more information, refer to Ref. [¹⁴] to find the definition of the appeared constants in the current subsection, or refer to Eq. (1) to find the values of the constant.

2) Mass flux term which is called G-term. G-term= [(c – dX_e + gX_e |X_e|)G + s] where G is the mass flux, and the other parameters are constant and/or later will be given in Eq. (1).

3) Thermo-hydraulics equivalent diameter of the sub-channel which is called D_h-term. The equivalent diameter is found based on

D_{\mathrm{h}}\mathrm{-term\; =}\left({c\prime }^{\mathrm{}}-{d\prime }^{\mathrm{}}{X}_{\mathrm{e}}\right)\left(j+k\exp (-m{D}_{\mathrm{h}})\right),

where c', d', j, and k are constants, D_h is the thermal-hydraulics heated diameter.

4) And finally the enthalpy rise in the channel which is shown by Δh-term. The enthalpy rise in the channel can be found by

Δh-term= u + v(h_f– h_in). (u and v are internal energy and specific volume, respectively.).

h(z)will be computed based on

h\left(z\right)=h_{\mathrm{in}}+{\displaystyle \frac{\mathrm{1}}{\dot{m}}}{\displaystyle {\int }_{-L/\mathrm{2}}^Z{q^{\prime }}_{\mathrm{rod}}}\left(z^{\prime }\right)\mathrm{d}{z}^{\prime },

where \dot{m} is the coolant flow rate in terms of kg/s.

For more details of the DNB computation, for a typical VVER-1000, please refer to Refs. [¹⁰, ¹²].

In this paper, an equation has been developed for computing {q^{″}}_{\mathrm{CHF}}, based on the W3 correlation, for axially uniform heated channel. This numerical equation is developed for a PWR core based on its inlet enthalpy, its local pressure, local quality, local equivalent diameter, and finally local water enthalpy as

(1)\begin{array}{l}{q^{″}}_{\mathrm{DNB-uniform}}=(P-\mathrm{term})\times (G-\mathrm{term})\times (D_{\mathrm{h}}-\mathrm{term})\times (\mathrm{\Delta }h-\mathrm{term})\hfill \\ =\left\{\right(\mathrm{2.022}-\mathrm{0.06238}P)\hfill \\ \\ +(\mathrm{0.1722}-\mathrm{0.01427}P)\exp \left[\right(\mathrm{18.177}-\mathrm{0.5987}P\left)X_{\mathrm{e}}\right]\left\}\right[(\mathrm{0.1484}-\mathrm{1.596}{X}_{\mathrm{e}}\hfill \\ +\mathrm{0.1729}{X}_{\mathrm{e}}\left|X_{\mathrm{e}}\right|)\mathrm{2.326}G+\mathrm{3271}](\mathrm{1.157}-\mathrm{0.869}{X}_{\mathrm{e}})[\mathrm{0.2664}\hfill \\ +\mathrm{0.8357}\exp (-\mathrm{124.1}{D}_{\mathrm{h}})\left]\right[\mathrm{0.8258}+\mathrm{0.0003413}(h_{\mathrm{f}}-h_{\mathrm{in}})],\hfill \end{array}

where {q^{″}}_{\mathrm{DNB-uniform}}={q^{″}}_{\mathrm{CHF-uniform}} is called the uniform critical heat flux, and is found in terms of kW/m²; P is the local pressure and should be between 5.5 MPa to 16 MPa; G is the mass flux of the most rated channel and should be between 1356 kg/(m²∙s^–1) to 6800 kg/(m²∙s^–1); X_e is the equilibrium quality and should be between −0.15 and 0.15; D_h is the thermal-hydraulics heated diameter and is between 0.015 m to 0.018 m; X_e is the equilibrium quality and should be between −0.15 to 0.15; L is the channel height and is between 0.254 m and 3.8 m; h_f is the water enthalpy in the corresponding vertical axis in term of kJ/kg, and h_in is the inlet enthalpy and should be>930 kJ/kg.

Equation (1) is developed for axially uniform heated channels only, and provides the value of the uniform CHF at any given axial location.

The axially non-uniform heated-flux is derived from {q^{″}}_{\mathrm{DNB-uniform}} with the correction factor F such as

(2){q^{″}}_{\mathrm{CHF}}\left(\mathrm{non-uniform}\right)={\displaystyle \frac{{q^{″}}_{\mathrm{CHF}}\left(\mathrm{uniform}\right)}{F}},

where F is a dimensionless factor, called the shape factor, and is found by

(3)F={\displaystyle \frac{c{\displaystyle {\int }_{\mathrm{0}}^l{q}^{″}\left(z^{\prime }\right)\cdot \exp [-C(l-z^{\prime }\left)\right]}\mathrm{d}{z}^{\prime }}{q^{″}\left(l\right)\cdot [\mathrm{1}-\exp (-Cl\left)\right]}},

where l is the length of the hottest channel that DNB to be calculated, and C is defined as (in terms of m⁻¹)

(4)C={\displaystyle \frac{\mathrm{4.23}\times {\mathrm{10}}^{\mathrm{6}}[\mathrm{1}-x_{\mathrm{e}}{\left(l\right)}^{\mathrm{7.9}}]}{G^{\mathrm{1.72}}}}.

Finally, the DNBR is found from Eq. (5),

(5)\mathrm{DNBR}={\displaystyle \frac{{q^{″}}_{\mathrm{CHF}}\left(\mathrm{non-uniform}\right)}{q^{″}\left(z\right)}}={\displaystyle \frac{{q^{″}}_{\mathrm{CHF}}\left(\mathrm{uniform}\right)}{F{q}^{″}\left(z\right)}},

{q^{″}}_{\mathrm{CHF}}\left(\mathrm{uniform}\right) and F are found based on Eqs. (1) and (3) where the obtained results are given in Table 4. The uniform critical heat flux varies among the vertical axis and is measured from the bottom of the core.

In addition, non-uniform critical heat flux and DNBR variations across the vertical channel are given in Fig. 6(a) and Fig. 6(b), respectively. Based on Fig. 6(b), it can be found that the MDNBR occurs at a vertical position of Z_DNB = 2.13 m in which at this point DNBR= 2.62 which is the value of MDNBR (Based on Final Safety Assessment Reports, MDNBR should not be less than 1.66 in the normal operation. More MDNBR makes more safety margins in the newly proposed TVS-2M core). This is a very precise result which confirms that DNB does not occur in the newly suggested TVS-2M core which is absolutely held in the safety margin for DNB concepts.

Maximum of fuel and clad temperatures

Temperature-dependent heat conduction

Experiments showed that the pellet (U+ Gd)O₂ heat conduction, k_f, is in the form of k_f = (a + bT)^–1 [¹⁴, ¹⁵] where a and b are constants, and T is temperature. Based on the temperature-dependent heat conduction coefficient (by solving the Fourier differential equation in which Fig. 7 gives the radial heat decreasing and all parameters defined in Fig. 7), it can be observed that

(6)T_{\mathrm{o}}-T_{\mathrm{s}}={\displaystyle \frac{\mathrm{1}}{b{k}_{\mathrm{f}}}}[\mathrm{1}-\exp ({\displaystyle \frac{b{q}^{\prime }}{\mathrm{4}\mathrm{\pi }}}\left)\right],

and

(7)T_{\mathrm{ci}}-T_{\mathrm{co}}={\displaystyle \frac{\mathrm{\Delta }c}{k_{\mathrm{c}}}}\frac{q^{\prime }}{\mathrm{2}\mathrm{\pi }R},

where Δc and k_c are the clad thickness and clad heat conduction coefficient, respectively.

Obviously, exp((bq'/4π) approaches to (1 – bq'/4π) for very small b, and the right hand side of Eq. (6) approaches to q'/(4πk_f) (i.e., approaches to temperature-independent heat conduction).

Temperature-independent heat conduction

Figure 8 shows the variations of heat conduction coefficient versus temperature for different contents of Gd [¹⁵, ¹⁶]. According to Fig. 8, the thermal conductivity decreases with the increase of the gadolinium oxide content (density) by a few percent.

Although the heat conduction is a function of temperature, in the desired fuel temperature range (up to 770 K to 1900 K), its variations is quantitatively small (c.f., Fig. 8, based on experimental values), and therefore its mean value is quite enough for the current estimation regarding the maximum temperature limits of fuel and clad.

Maximum fuel and clad temperatures for LWRs and for different types of fuel (in the case of constant pellet heat conduction, k_f which is a constant, are summarized as [¹⁷]:

1) Solid Pin: The maximum radial temperature (T_max) occurs in the center of the pellet of fuel, and is equal to (and also axially maximized at mid-plate of the core, at z = 0):

(8)T_{\mathrm{o}}=T_{\max }=T_{\mathrm{s}}+{\displaystyle \frac{{q^{\prime }}_{\max }}{\mathrm{4}\mathrm{\pi }{k}_{\mathrm{f}}}},

where T_s and {q^{\prime }}_{\max } are outer surface pellet temperature and the maximum linear heat rate, respectively. Thus, a limit on the {q^{\prime }}_{\max } directly implies a design requirement on the maximum fuel temperature.

2) Pin with central annular hole: The maximum radial temperature for an annular central hole pellet, and for a constant heat conduction is

(9)T_{\mathrm{o}}=T_{\max }=T_{\mathrm{s}}+{\displaystyle \frac{{q^{\prime }}_{\max }}{\mathrm{4}\mathrm{\pi }{k}_{\mathrm{f}}}}[\mathrm{1}-{\displaystyle \frac{\ln {(R_{\mathrm{o}}/R_{\mathrm{i}})}^{\mathrm{2}}}{{(R_{\mathrm{o}}/R_{\mathrm{i}})}^{\mathrm{2}}-\mathrm{1}}}],

where R_o and R_i are the outer and inner radii of fuel, respectively. This type of the pin is used in the VVERs reactors which is a topic of the present research, and will be explained later in this paper in more detail as well as the maximum temperature of the clad for this fuel type.

3) Annular pellet with internally and externally cooling: The steady-state Fourier heat conduction differential equation has been solved with the assumption that the temperature is only a function of r, due to symmetry, and for constant k_f:

(10){\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}\left(r{\displaystyle \frac{\mathrm{d}T\left(r\right)}{\mathrm{d}r}}\right)+{\displaystyle \frac{q^{‴}r}{k_{\mathrm{f}}}}=\mathrm{0.}

Taking two times integration, it is found that

(11)T\left(r\right)=-{\displaystyle \frac{q^{‴}{r}^{\mathrm{2}}}{\mathrm{4}{k}_{\mathrm{f}}}}+C_{\mathrm{1}}\ln \left(r\right)+C_{\mathrm{2}}.

But, in this type of fuel, the boundary values are T (r = R_fi) = T_s (inner radius of fuel), and T (r = R_fo) = T_s (outer radius of fuel). Note that the surface temperatures for the outer and inner radii of fuel are the same here, and it is observed that the radial temperature distribution is

(12)T\left(r\right)=T_{\mathrm{s}}+{\displaystyle \frac{q^{\prime }}{\mathrm{4}\mathrm{\pi }{k}_{\mathrm{f}}}}[R_{\mathrm{fi}}^{\mathrm{2}}-r^{\mathrm{2}}+{\displaystyle \frac{R_{\mathrm{fo}}^{\mathrm{2}}-R_{\mathrm{fi}}^{\mathrm{2}}}{\ln (R_{\mathrm{fo}}/R_{\mathrm{fi}})}}\ln (r/R_{\mathrm{fi}}\left)\right].

Clearly, by inserting its derivative equal to zero, the radial dimension at which temperature has a maximum value, r_max, can be obtained by

(13)r_{\max }=\sqrt{{\displaystyle \frac{R_{\mathrm{fo}}^{\mathrm{2}}-R_{\mathrm{fi}}^{\mathrm{2}}}{\ln {(R_{\mathrm{fo}}/R_{\mathrm{fi}})}^{\mathrm{2}}}}}.

Finally, substituting Eq. (13) into Eq. (12), and by considering q' = q'''π(R_{\mathrm{fo}}^{\mathrm{2}}-R_{\mathrm{fi}}^{\mathrm{2}}), the result is

(14)T_{\mathrm{o}}=T_{\max }=T_{\mathrm{s}}+{\displaystyle \frac{{q^{\prime }}_{\max }}{\mathrm{4}\mathrm{\pi }{k}_{\mathrm{f}}}}[{\displaystyle \frac{R_{\mathrm{fi}}^{\mathrm{2}}-r_{\max }^{\mathrm{2}}}{R_{\mathrm{fo}}^{\mathrm{2}}-R_{\mathrm{fi}}^{\mathrm{2}}}}+{\displaystyle \frac{\ln (r_{\max }/R_{\mathrm{fi}})}{\ln (R_{\mathrm{fo}}/R_{\mathrm{fi}})}}].

Now, let us estimate the order of magnitudes of T_o for the above mentioned three cases. By focusing on Fig. 8, a mean constant thermal conductivity of (92%U+ 8%Gd)O₂ is assumed to be k_{\mathrm{f}}=\mathrm{2.65}{\displaystyle \frac{w}{m-k}} , in the desired fuel temperature range of operation. As it can be seen in Fig. 8, the experimental temperature range is within 770–1080 K. By extrapolating the experimental values using linear extrapolation, the desired temperature range (770–1900 K) can be found in which fuel radially operates. For (U+ Gd)O₂ pellet, the surface temperature of T_s = 700 K (at approximately Z_DMB = 0, near the mid-plane of the core so that q''' is maximized) is a value close to the real fuel rod which operates in the core; and the maximum linear heat rate of q'_max = 44.8 kW/m is the upper limits for a PWR; it is found that the maximum fuel temperatures are 1680^oC (for solid pin), 1656^oC (for central hole type pin), and 845^oC (for annular pin). For the mentioned estimations, the real dimensions of the pellet have been used which are:

For solid pin: R = 4.1 mm; For central annular hole pin: R_fo = 3.8 mm and R_fi = 0.5 mm; For internally and externally cooling pin: R_fo = 7.05 mm and R_fi = 4.85 mm.

As it can be seen, at the same linear heat rate, the same outer fuel temperature, and the same heat conduction coefficient, the externally-internally cooling annular fuel has the least maximum temperature in its center, and it is a unique result that causes more safety operation but which is out of the scope of the current research. For more details, please refer to Refs. [¹⁷, ¹⁸]. Obviously, at the same linear heat rate, the outer fuel temperature, and the same heat conduction coefficient, the maximum central temperatures of fuel type (ii) are less than that of the fuel type (i), and it is a unique result which causes more safety.

After estimating the maximum radial temperature for type (ii) which is our favorite in the current research (This maximum occurs in the center of pellet and for z = 0. For estimating its mean value across the vertical direction, q'_ave = 16 kW/m should be inserted), the maximum fuel and clad temperature of the new TVS-2M pins should be determined and its upper limits be found. As it has been explained earlier, the material of each fuel pin is a mixture of gadolinium and uranium oxides with specific enrichments as shown in Fig. 2, and the desired data which are used in the current subsection are given in Table 5.

It is well-known that the maximum axial clad temperature (which is a safety margin) occurs upper than the mid-plate (c.f., Eq. (22) by inserting specific values). Specifically, to find the axial temperature distributions, consider Fig. 9.

Suppose a fuel rod with a length of l where its neutronics extra polated length is l_ex as shown in Fig. 9. The axial volumetric heat rate approximately varies as

(15)q^{‴}={q^{‴}}_{\max }\cos \left({\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}}\right).

For a differential element of dx at a distance x from the center of the coordinate (mid-plane), an energy balance can be written between fuel element and coolant (It has been assumed that all fuel fission energy is transferred to coolant after passing through gap and clad thickness radially, and therefore omitted any axial energy transferring).

(16)\dot{m}{C}_{\mathrm{P}}\mathrm{d}{T}_{\mathrm{f}}\left(x\right)=q^{‴}A\mathrm{d}x,

where \dot{m} is the coolant mass flow rate, and C_P is the specific heat at constant pressure. By integration,

(17)\dot{m}{C}_{\mathrm{P}}{\displaystyle {\int }_{T_{\mathrm{f1}}}^{T_{\mathrm{f}}}\mathrm{d}{T}_{\mathrm{fluid}}}={q^{‴}}_{\max }A{\displaystyle {\int }_{-\frac{l}{\mathrm{2}}}^x\cos \left({\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}}\right)\mathrm{d}x}.

Therefore, the axial fluid (water) temperature is

(18)T_{\mathrm{fluid}}\left(x\right)=T_{\mathrm{f1}}+{\displaystyle \frac{A{q^{‴}}_{\max }{l}_{\mathrm{ex}}}{C_{\mathrm{P}}\mathrm{\pi }}}(\sin ({\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}})+\sin ({\displaystyle \frac{\mathrm{\pi }l}{\mathrm{2}{l}_{\mathrm{ex}}}}\left)\right),

where T_f1 is the inlet temperature. Now, suppose all heat energy is transferred from fuel differential element to gap and clad differential elements.

(19){q^{‴}}_{\max }A\cos {\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}}\mathrm{d}x=h_{\mathrm{o}}\left(\mathrm{2}\mathrm{\pi }{r}_{\mathrm{o}}\right)\left[T_{\mathrm{o}}\right(x)-T_{\mathrm{fluid}}(x\left)\right]\mathrm{d}x,

where h_o is the gap convection coefficient at position x, and T_o(x) is the clad temperature at x. Substituting T_fluid(x) from Eq. (18) into Eq. (19), it is found that the axial variation of the clad temperature is given by

(20)T_{\mathrm{o}}\left(x\right)=T_{\mathrm{f1}}+{\displaystyle \frac{{q^{‴}}_{\max }A{l}_{\mathrm{ex}}}{C_{\mathrm{P}}\mathrm{\pi }}}(\sin {\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}}+\sin {\displaystyle \frac{\mathrm{\pi }l}{\mathrm{2}{l}_{\mathrm{ex}}}})+{\displaystyle \frac{{q^{‴}}_{\max }A\cos (\mathrm{\pi }x/l_{\mathrm{ex}})}{\mathrm{2}\mathrm{\pi }{r}_{\mathrm{o}}{h}_{\mathrm{o}}}}.

Therefore, the clad temperature gradient and position at which the clad has maximum temperature, respectively are

{\displaystyle \frac{\mathrm{d}{T}_{\mathrm{o}}\left(x\right)}{\mathrm{d}x}}=\frac{{q^{‴}}_{\max }A}{C_{\mathrm{P}}}\cos {\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}}-\frac{{q^{‴}}_{\max }A}{\mathrm{2}\mathrm{\pi }{r}_{\mathrm{o}}{h}_{\mathrm{o}}}\sin {\displaystyle \frac{\mathrm{\pi }x}{l_{\mathrm{ex}}}},

(21)q^{\prime }=q^{‴}\mathrm{\pi (}{R}_{\mathrm{fo}}^{\mathrm{2}}-R_{\mathrm{fi}}^{\mathrm{2}}\mathrm{),}

(22)l_{\mathrm{o}-\max }={\displaystyle \frac{l_{\mathrm{ex}}}{\mathrm{\pi }}}{\tan }^{-\mathrm{1}}{\displaystyle \frac{\mathrm{2}{l}_{\mathrm{ex}}{r}_{\mathrm{o}}{h}_{\mathrm{o}}}{C_{\mathrm{P}}}}.

If axial active fission length is divided into 10 grids, a position between grids 6 and 7 is approximately the location of the maximum clad temperature (Eq. (22)) along vertical axis which is shown in Fig. 9. Table 5 gives all the required data for the current computations, and the value of the maximum clad temperature (at normal operation and for constant C_P) is found to be about 346^oC, which is a safe margin to avoid clad (zircaloy-4+ 1%Nb) melting. It is very important to emphasize here that a little change of C_P occurs due to pressure drop across the vertical axis. So, it has been assumed that an excess temperature of 10^oC for the clad based on a little variation of C_P, and hence the upper limit of the maximum clad temperature does not exceed 356^oC which is still far from the clad melting point.

By considering Eq. (18), it can be found that C_P is the specific heat at constant pressure. By considering C_P = constant for the PWR (at pressure of 15.7 MPa), C_P can be brought out from integral of Eq. (18). Therefore, the final Eqs. (21) and (22) are derived from C_P = constant.

In a real VVER (or other PWRs), a minor change of core pressure exists. Numerically, about 30–60 kPa pressure drop exists across the top and bottom of the core. This pressure variation (60 kPa) in comparison to 15.7 MPa is very small and does not strongly affect the calculation of maximum axial clad temperature. Table 4 also gives the theoretical data of pressure drop across the axial direction of the core. As it can be seen, the enthalpy decreasing is very small.

Based on the above explanations, it is assumed that considering a variable (fluid temperature dependent) C_P does not affect the estimation of the maximum clad temperature. At maximum, variable C_P causes 10^oC more than the calculated value (It is a very extreme value). This extreme maximum clad temperature would be 356^oC which is still far away from the melting point of the clad. Actually, 10^oC is considered as a very extreme value if C_P is not constant. 10^oC is thought to be a very extreme value, but 5^oC is not extreme enough. Besides, 15^oC would not be a real scientific assumption based on the above small pressure drop across the core.

Core neutronics margins and their limits

For computing the neutronics margins and the control rod reactivity margins, those are other important safety margins for every PWR, and are mentioned in Table 3.11 in Ref. [⁶]. The WIMS code has been used for core modeling. To find an example, the WIMS input file for an optional case is given in Appendix. Moreover, the PARCS code is used to cover the control rod motions and their transient effects. First of all, a short explanation regarding the implementation of the PARCS code will be given. For more details of the WIMS code, please refer to Ref. [¹⁸].

Implementation of PARCS and SARCS codes

The PARCS and SARCS codes are used for transient conditions regarding the control rod insertion (or withdrawn), burn up calculations, and so on. Since the PARCS code [⁷] is not able to get the required data directly from the WIMS-D5 cell calculation code (Which were used in the hole core configuration for criticality approach, and other neutronics behaviors which will appear in the upcoming sections.), an interface-code called SARCS is used. The SARCS code is developed in Shiraz University [⁷]. The PARCS input data should be in the specified format called PMAXS, and SARCS brings out the WIMS-D5 output into the PMAXS format to be used as the input file of the PARCS code. The specific trends of the SARCS execution and details of implementation of the PARCS code are shown in Fig. 10.

The flow chart descriptions are as follows:

1) The left hand side:

(1) State variable insertion: Insert the input data such as fuel temperature, coolant temperature, boric acid concentration, banks position of control rods, and moderator density.

(2) Set branches and burnups: Each branch is defined as the combination of mentioned state variables which define a specific state of reactor operation (combination of the five States variables are called a “branch,” c.f., below to find more).

(3) Prepare the WIMS input file for each branch.

(4) Extract required data (group diffusion coefficient as well as group cross sections) from the output of WIMS to construct PMAXS format.

(5) Last burn up iteration: This iteration follows to obtain all materials group constants for a specified burnup rate (a unique WIMS input is given in the Appendix).

(6) Construct the PMAXS using SARCS interface program [⁷].

2) The right hand side:

(1) Call PMAXS subroutine: the PMAX format file is read as a cross sectional file format to be the input file of the PARCS.

(2) Node-wised cross sections: in PARCS code each fuel assembly is known as “node,” and read data of each node from the PMAX data.

(3) Neutronics calculations: the neutronics calculations are conducted for each node.

(4) Node-wised power: the power data are obtained for each node.

(5) T-H module: the thermal-hydraulics calculations are conducted for each node.

(6) T-H state: feedbacks of the T-H are inserted into each node, based on the T-H calculations, and the new PMAXS input are obtained based on the current T-H feedbacks.

(7) The new data from burnup calculations of the right hand side are inserted (fuel depletion module).

(8) History for each region: fuel depletion and burnup calculations of the current iteration are found from the last iteration step.

Production of the cross-sections in the PMAXS format are achieved on the basis of GENPMAX manual [¹⁹, ²⁰]. The SARCS interface program must be corrected for the TVS-2M core and the new library data should be generated. Based on Table 6, for different types of FAs, the input codes of WIMS-D5 are prepared for state conditions of CZP, HZP, and HFP in which their physical differences were explained in Subsection 3.1. It should be noted here that the SARCS considers five sets of variables, including the control rod (CPS-CR), coolant density (CD), the concentration of soluble absorber (PC), the temperature of fuel (TF), and the temperature of coolant (TC), as the main states variables in its calculation. Then, SARCS prepares the PMAXS library for each fuel assembly using execution of the WIMS-D5 for different mentioned states.

Values of the five different state variables for different conditions are given in Tables 7 to 10. The combination of these five state variables is called a “branch.” A branch has three modes which are called the reference mode, the basic mode, and the current mode.

According to the initial conditions, the reference mode is defined. The current mode is found based on the change of a variable. SARCS finds the basic branch for any current mode, and performs partial cross section calculations. For more details, please refer to the GENPMAX manual and Refs. [¹⁹, ²⁰].

Control rods shut-down margin, integral and differential worth, and most RCCA

PARCS benchmarking for the TVS-2M core

One of the most efficient safety margins is the control rod shut-down margin as an important safety limit of each core. The results for the TVS-2M core are obtained based on the infinite multiplication factor approach in the WIMS-D5 and PARCS codes which are tabulated and compared in Table 10. In other words, Table 10 shows the infinite multiplication values for each FA of the TVS-2M core at HZP. These calculations are carried out for PARCS benchmarking (code to code benchmarking to find accuracy of the PMAX inputs for the TVS-2M core) with the world-wide known WIMS code.

Integral and differential worth, and some applied operational criteria for the CPS-CRs movements

After the above successful benchmarking, the differential and integral worth (of the TVS-2M as well as current TVS cores) for the most RCCA called No. 8, No. 9, and No.10 control banks are found, and compared. In Figs. 11, 12, and 13, the values of the integral and differential worth and the concentrations of the boric acid during insertion of the control banks (for both cores) are shown (new core= TVS-2M, and current core= TVS). It should be mentioned here that, distributions of the CPS-CRs for the TVS and TVS-2M cores are given in Figs. 3(a) and 3(b), respectively. Based on Fig. 3, the numbers of all CPS-CRs groups are 85 and 103 for the TVS and TVS-2M, respectively. Besides, the total length of the CPS-CRs for the TVS and TVS-2M cores are equal to 353 cm and 368 cm, respectively. Moreover, the total reactivity of the control rod number 10 is equal to 0.8b, and the speed of control rod (inserting and withdrawing) is equal to 2 cm/s in a normal condition.

A very important rule in all VVERs is that, the primary circuit follows the secondary loop (called turbine mode). Therefore, the core is regulated by the turbine demand, but for neutronics decreasing and increasing there are two possible ways:

1) If there is enough time: boric acid insertion (or oppositely water insertion) is a safe method for neutron decreasing (or increasing).

2) If there is not enough time: inserting (withdrawing) control rods 8, 9, and 10 are the methods for neutronics decreasing (increasing).

In other words, the CPS-CRs 8 and 9 are used for major power increasing (decreasing) and for off-setting the poison build up effects (e.g., Xe). It is also noticed here that the control rods No. 1 to No. 7 (c.f., Fig. 3) are not used for power increasing (decreasing), and are withdrawn from the core all the cycle time. Those are used only for the emergency conditions as the emergency-dropping control rod together with the CPS-CRs 8, 9, and 10.

It should be emphasized here that, by focusing on Figs. 11 and 12, it is found that the control rod No. 8 is the most reactive control rod for both cores, and control banks No. 9–10 are the second-third worth, respectively. It is a very noticeable result which is confirmed by a similar Russian reactor called Rostove NPP [²¹].

Axial and radial power peaking factors

The axial and radial power peaking factors are another major design limits. For their finding in the PARCS, it has been proposed that the power-flux parameters are defines as: N: the current reactor power; N_nom: the nominal reactor power (3000 MW); N_perm: the maximally permissible reactor power, at the current state of the equipment; K_qj: the relative power of jth FA which is equal to its power to average FA power; Kv_ij: the core peaking factor in a cell (i, j) of the core (i is the height node number, and j is the FA number.) which is equal to the power-flux ratio in this cell to the average power; Kv_ijperm: maximally permissible Kv_ij value at power N_perm; and Ql_ij: the maximum linear heat rate in the core cell (i,j) in terms of W/cm.

By focusing on Fig. 14, it is found that the maximum of relative power of FAs peaking factor (K_q) is near 1.6 for the TVS-2M core which is an acceptable limit.

Because an accepted and verified condition for K_q is given by K_q<K_q-max× N_a/N_p where N_a is the maximum power (100%) and N_p is the current power. For instance, when a core is held at 35% power, K_q<1.6 × 100/35, and then K_q<3.85. Therefore, the maximum calculated K_q = 1.6 is an acceptable limit [²²].

Moreover, by focusing on Fig. 14, it is found that the peaking factor of TVS-2M FAs is more than the current FAs, and it is a noticeable result. It is very important to emphasize that for the first 50 d.

Besides, the maximum linear heat rate during the cycle is given by Fig. 15. It is emphasized that the maximum linear heat rate is within the safe margin (below maximum value of 45 kW/m for a PWR). Basically, the axial variations of the linear heat rate are approximately given by (in an arbitrary time or burn up, and in the case of a homogeneous FA)

(23)q^{\prime }={q^{\prime }}_{\max }\sin \left({\displaystyle \frac{\mathrm{\pi }{Z}_{\mathrm{DNB}}}{H}}\right)=\mathrm{410}\sin \left({\displaystyle \frac{\mathrm{\pi }{Z}_{\mathrm{DNB}}}{H}}\right)\hspace{0.17em}\mathrm{W/cm},

where Z_DNB is measured from the bottom of the core, H is the extrapolated fuel active length, and q'_max = 410 W/cm. The average value of Eq. (23) is less than 171 W/cm which is a very good margin (i.e., q'_ave = 171 W/cm). In addition, based on Fig. 15, it is seen that q'_max for the TVS-2M core is less than the current TVS core, and it makes more safety margin for the future advanced TVS-2M core. The average linear heat rate for the current core (TVS core) = 167.64 W/cm and the average linear heat rate for new core (TVS-2M core) = 160.29 W/cm.

For comparison, the APPF during the cycle is shown in Fig. 16. As it is seen, its maximum is near 1.6 (and more than the current core for the first 30 d), but it is also within the safety margin.

Control rod withdrawn accident (most RCCA withdrawn) as a standard DBA

In the current subsection, the most reactive control rods withdrawn at HFP are evaluated so that it is a familiar DBA scenario. To construct the mentioned scenario, it is supposed that at time zero, components of all plant are in the normal working conditions (such as SGs, turbine, feed-water, core, etc.), and all components and parameters of the primary and secondary loops are held in their authorized level. Then, in 0.1 s, the most reactive control rods (e.g., bank No. 8) suddenly fully drop in the core. As shown in Fig. 17, a major negative reactivity is inserted due to the drop-rod accident, and therefore the relative thermal flux extremely decreases and the output energy which is going to the SG is suddenly decreased. But, at the same time, in the secondary loop, the turbine is working at normal rated full power and this abnormality (decreasing inlet temperature of water into SG from the primary loop which is suddenly changed, and at the same time, the fixed outlet steam from SG to the turbine at the secondary side), reduces the average pressure in the SG (average pressure is formed by inlet water and outlet steam in the SG) which causes the outlet water temperature of the SG to decrease. Based on the mentioned abnormality in the SG, inlet water to the core (at the primary loop) will be decreased, which causes the average temperature of the core to decrease (versus before the drop-rod accident). By increasing the water density (or due to negative temperature coefficient a_T = ∂r⁄∂T_water) reactivity will be increased (∂r should be increased when ∂T_water is decreased to maintain negative a_T.). Therefore, more thermal power should be obtained. In other words, a negative coolant temperature feedback makes a worse situation here. This abnormal increasing of thermal power may cause a local hot-spot and local excess temperature more than that of the clad melting point which causes a damage in a few fuel elements. If the thermal power reaches more than 107% and/or the core period becomes less than 10 s, the scram signal is sent. The major controlling systems are based on the power and the period. If the core thermal power exceeds 107%, the power controlling sensors order scram signals (or called emergency shut-down signals). In addition, if the core period becomes less than 10 s (This means that the power would be e = 2.73 times during the next 10s, and in the next 10 s would be e² times, and so on), the scram signals will be ordered [²³]. Figure 17 shows the reactivity variations a few seconds after full-insertion of the most reactive control rods at different initial powers of 1%, 70%, and 105% (series 3).

Boric acid concentration during the first-cycle

Figure 18 shows the change of the boric acid concentrations during the first cycle for the current and new TVS-2M cores. As is known, a near-linear boric acid decreasing occurs to maintain the uranium depletions and the holding approach of criticality. It is found that, the new core TVS-2M is working with more boric acid concentrations than the TVS core at the same life-cycle. It is a negative result which causes more expenses for core management.

Calculations of reactivity coefficients

Reactivity coefficients including Doppler and moderator coefficients are very important and sufficient data for all operating reactors as mentioned before in Subsections 3.1 and 3.2. The fuel temperature coefficient of reactivity, the boron concentration, and the water density coefficient of reactivity during the first cycle are evaluated (both cores for comparison), and are given in Figs. 19, 20, and 21, respectively.

To find the moderator temperature coefficient of reactivity, a step size of temperature variation of 5^oC for the coolant is (theoretically not reality) considered, and the effective multiplication factor for the current step-size is found [²³, ²⁴]. It is known that, by coolant temperature increasing, the density of water will decrease. Then, the corresponding reactivity of the current step-size coolant temperature is found. By plotting reactivity versus coolant temperature, and finding the slop of the curve, the coolant reactivity coefficient is evaluated. These procedures are continued at another burnup (fuel cycle) from the fuel depletion data (c.f., 4.7). The same procedures are performed for calculating the Doppler coefficient, but with a step size assumption of 50^oC. As it is seen in Fig. 19, the Doppler reactivity coefficient is close to −2.5 ×10⁻³∆k/k/°C.

There are some fluctuations in the fuel reactivity coefficient during the cycle-time (or by increasing the burnup), specifically for the TVS-2M core which contains mixed (U+ Gd)O₂ fuel. Basically, by increasing the burnup, the fuel temperature coefficient should be slightly increased and this physical phenomenon is seen for both cores. Because, based on Fig. 19, it is seen that the trend of the values of the fuel temperature coefficient is upward during the life-cycle. But, actually there is not any physical interpretation regarding some perturbations during the life-cycle, especially for the TVS-2M core. It must be emphasized that, it may be caused by the calculational errors, and the suggested cross sections modification.

Absolutely, the boric acid reactivity coefficient during the cycle corresponds to the variations of water density. By decreasing the boric acid in the cycle, the boric acid reactivity coefficient for the TVS-2M will be more decreased (c.f., Fig. 20). It is emphasized here that each reactivity coefficient is calculated independently. For instance, to compute the boric acid coefficient, the coolant and fuel coefficient of the reactivity are ignored to find only the contribution of the boric acid coefficient. The same procedures are performed for the other reactivity coefficients.

Finally, Fig. 21 shows the water density reactivity coefficient which corresponds to the boric acid coefficient. From Fig. 20 it is seen that the boric acid reactivity coefficient decreases specially, for the TVS-2M core, during the life-time (their trend-lines) which is confirmed in any fundamental text books [⁶]

First cycle burnup estimations

The burnup estimations are found based on the linear reactivity [²⁵]. Based on the linear reactivity model, burnup= 9.5 × 10⁵w can be obtained where w is given by fuel depletion analysis as

(24)w={\displaystyle \frac{M_{\mathrm{f}}\left(\mathrm{235}\right)+M_{\mathrm{f}}\left(\mathrm{238}\right)+M_{\mathrm{f}}\left(\mathrm{249}\right)+M_{\mathrm{f}}\left(\mathrm{241}\right)}{M_{\mathrm{235}}^{\mathrm{0}}+M_{\mathrm{238}}^{\mathrm{0}}}}.

In which M_f(235), M_f(238), M_f(249), and M_f(241) are the fission fragments which are obtained from U-235, U-238, Pu-239, and Pu-241 fissioning. In addition, M_{\mathrm{235}}^{\mathrm{0}} and M_{\mathrm{238}}^{\mathrm{0}} are the initial contents of total uranium isotopes (at t = 0).

Now suppose that a thermal reactor of power of 3000 MW_th (1000 MWe) is operated with an availability of 0.8. From the fuel depletion data, the numerator is found at an arbitrary time. Moreover, the initial contents of uranium in the TVS-2M and TVS cores are known (c.f., Subsection 3.1). Finally, Fig. 22 gives the results for both cores. As it can be seen, the burnup of the TVS-2M core is more than the TVS core (1.18 MWd/kgU more) which corresponds to 45.6 more days of cycle length.

It is very important to emphasize here that, the first cycle burnup for the TVS-2M core is 13.7 MWd/kgU (based on Fig. 22), which is one of the most important improvements of the TVS-2M core versus the current TVS core in the future advanced VVERs.

In another word, based on the calculations (Which are briefly explained in the above sentences.), the first cycle burnup of the TVS-2M core has been 13.7 MWd/kgU. But the first cycle burnup for the TVS core is 12.75 MWd/kgU. Therefore, an extra 1.18 MWd/kgU will be achieved in the TVS-2M core, which is a goal for upcoming VVERs.

Conclusions

This paper which is an independent study on the Russian newly proposed FAs called TVS-2M, would be applied for the future advanced VVERs. Some important aspects of the neutronics and thermal hydraulics investigations (and analysis) of the mentioned TVS-2M FAs are conducted, and the results are compared with the standards PWR CDBL. The obtained neutronics and/or safety features are classified as follows:

1) Control rods shut-down margin, integral and differential worth, and most RCCA;

2) The axial and radial power peaking factors;

3) The control rod withdrawn accident scenario;

4) The boric acid concentration during the first-cycle;

5) The reactivity coefficients including moderator, Doppler, water density, and boric acid; and

6) The discharge burnup in the first-cycle.

The above parameters are benchmarked with the CDBL of a PWR and improvement details (or may be a minor weakness) are investigated herein.

One of the main improvements of the newly proposed TVS-2M core is its more discharge burnup. The burnup calculations indicate that ~ 46 d increase of the first cycle-length (which corresponds to 1.18 MW_thd/kgU increase of burnup) may be achieved, which may be the best improving aim of the new FA type. In addition, the investigations suggest that more intrinsic safety margins (Such as Doppler coefficient, the maximum linear heat rate, and the average linear heat rate of the TVS-2M core are less than maximum linear heat rate and average linear heat rate of the current TVS core.) are achieved in the new type of FA.

Moreover, one of the most important thermal-hydraulics CDBL such as MDNBR, and the maximum temperatures of fuel and clad (radially and axially) are calculated. The obtained results are also discussed according to design criteria of a PWR.

The current investigations on the TVS-2M core indicate that many CDBL are satisfied, and this new type of the core is verified as the future advanced VVERs. One of the most important progress of the TVS-2M core, (as an extra-improvement) would also be its more core intrinsic safety margins which are mentioned herein as well as more cycle burnups.

References

[1] Bibilashvili Yu K, Kuleshov A V, Mikheev E N, Investigation of thermal-physical and mechanical properties of uranium-gadolinium oxide fuel. In: Proceedings of IAEA Technical Committee Meeting on Improved Fuel Pellet Materials and Designs. Brussels, Belgium, 2003, 85–100

[2] Une K. Thermal expansion of UO₂-Gd₂O₃ fuel pellets. Journal of Nuclear Science and Technology, 1986, 23(11): 1020–1022

[3] Lysikov A V, Kouleshova V, Novikova E A. Results of thermal-physical and mechanical property investigations of modified uranium-gadolinium oxide doped fuel. In: Russian Scientific Conference Materials for Nuclear Technics. Zvenigorod, Russia, 2007, 22–44

[4] Nivikov V, Dolgov A, Molchanov V. In: TopFUEL Conference. Wurzburg, Germany, 2003

[5] Kosourov E, Oleksuk D, Pavlovichev A, Shecherenko A. Optimization of gadolinium distribution in FAs for WWER-1000. In: 10th International Conference on WWER Performance, Modelling and Experimental Support. Sandanski, Bulgaria, 2013

[6] Oka Y, Madarame H, Uesaka M. An advanced course in nuclear engineering: nuclear reactor design. Japan: Springer Press, 2014

[7] Porhemmat M H, Hadad K, Faghihi F. PARCS cross-section library generator; part one: development and verification. Progress in Nuclear Energy, 2015, 78: 155–162

[8] Safaei Arshi S, Mirvakili S M, Faghihi F. Modified COBRA-EN code to investigate thermal-hydraulics analysis of the Iranian VVER-1000 core. Progress in Nuclear Energy, 2010, 52(6): 589–595

[9] Lunin G L, Novikov A N, Pavlov V I, Pavlovichev A M. Development of four-year fuel cycle based on the advanced fuel assembly with uranium-gadolinium fuel and it simple mentation to the operating VVER-1000 units. In: Proceeding of the 10th Symposium of AER. Moscow, Russia, 2000, 1:75–95

[10] Amin Mozafari M, Faghihi F. Design of annular fuels for a typical VVER-1000 core: neutronic investigation, pitch optimization and MDNBR calculation. Annals of Nuclear Energy, 2013, 60: 226–234

[11] Mirvakili S M, Faghihi F, Khalafi H. Developing a computational tool for predicting physical parameters of a typical VVER-1000 core based on artificial neural network. Annals of Nuclear Energy, 2012, 50: 82–93

[12] Bagheri S, Faghihi F, Nematollahi M R, Behzadinejad B. Assessment of thermal hydraulics parameters of the VVER-1000 during transient conditions. International Journal of Hydrogen Energy, 2016, 41(17): 7103–7111

[13] Lamarsh J R. Introduction to Nuclear Reactor Theory. New York University: Addison Wesley Publishing Company, 1966

[14] Tong L S, Weisman J. Thermal Analysis of Pressurized Water Reactors. United States Department of Energy, Office of Scientific and Technical Information, OSTI Identifier: 5856843, 1979

[15] Novikov V V, Mikheev E N, Lysikov A V, Property investigations of (U,Gd)O₂ fuel with higher content of gadolinium oxide.

[16] Balestrieri D. A study of the UO₂/Gd₂O₃ composite fuel, IAEA Series.

[17] Todreas N E, Kazimi M S. Nuclear Systems I: Thermal Hydraulics Fundamentals. CRC Press, 2001

[18] Faghihi F, Fadaie A H, Sayareh R. Reactivity coefficients simulation of the Iranian VVER-1000 nuclear reactor using WIMS and CITATION codes. Progress in Nuclear Energy, 2007, 49(1): 68–78

[19] Downar T, Xu Y, Seker V, PARCS v2.7 US NRC Core Neutronics Simulator User Manual. Purdue University, West Lafayette, Ind, USA, 2006

[20] Downar T, Xu Y, Seker V, PARCS v3 US NRC Core Neutronics Simulator User Manual. University of Michigan, Ann Arbor, MI, USA, 2010

[21] Lunin G, Novikov A, Pavlov V, Advanced fuel cycles of VVER-1000 reactors. In: 8th International Conference on WWER Fuel Performance, Modelling and Experimental Support. Helena Resort near Burgas, Bulgaria, 2003

[22] Atomic Energy Organization of Iran. Album of Neutron and Physical Characteristics of the 1st Loading of Boushehr Nuclear Power Plant. BNPP FSAR, 2003

[23] Faghihi F, Saidi nezhad M. Two safety coefficients for a typical 13 × 13 annular fuel assembly. Progress in Nuclear Energy, 2011, 53(3): 250–254

[24] Erfani Nia A, Faghihi F, Hadad K. Prompt and power reactivity coefficients for the next generation VVER-1000 reactor including hexagonal assemblies and annular fuels. Progress in Nuclear Energy, 2012, 61: 41–47

[25] Driscoll M J, Downar T J, Pilat E E. The Linear Reactivity Model for Nuclear Fuel Management. La Grange Park, Illinois: American Nuclear Society, 1990

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