Effect of Radial Neutron Reflector on the Characteristics of Nuclear Fuel Burn-Up Wave in a Fast Neutron Energy Spectrum Multiplying Medium: A Consistent Parametric Approach

Most of the developed and developing countries are currently concentrating their focus on finding alternative energy sources to meet the demands prior to population growth. Nuclear energy provides a viable option as it is reliable, clean, and environment friendly. However, three major accidents, that is, Three-Mile Island (TMI) in 1979, Chernobyl in 1986, and the Fukushima accident in 2011, heightened concerns about the reliability and safety of nuclear energy. In the aftermath of these accidents, the priority of the reactor design is shifted toward component integrity, intrinsic safety, and long-term decay heat removal. Another primary concern is related to effective fuel storage. During the enrichment process of natural uranium, a small portion of it is converted into enriched fuel, while the rest of the material is stored as depleted uranium. The storage of depleted uranium raises the concern of nuclear proliferation as it can be used for the development of weapons. Researchers are rigorously working on the improvement of the safety and design of nuclear reactors. Breed and burn (B&B) concept is one of the promising ones among the other proposed concepts due to its inherent safety features. Accidents caused by uncontrolled chain reaction are improbable owing to the physical principles of this reactor design. Other advantage involves using natural or even depleted uranium as fertile fuel, which significantly lessens the requirement for fuel enrichment and spent fuel storage. These reactors can self-regulate and burn like a candle for several years. Unique phenomena like this provide substantial benefits such as the elimination of frequent fuel handling and removal of control rod mechanism compared to the other conventional reactors.

Feinberg [1] first postulated the possibility of propagation of self-sustained breed and burn wave in fissile material. Later, Feoktistov [2–4] confirmed the concept for U–Pu fuel cycle. Teller et al. [5,6] discussed the overall design of a future Th–U fueled solution fast reactor using Monte Carlo-based techniques, although the details of the mathematical modeling were sparsely discussed. Seifritz [7] provided an analytical model of burn-up wave in a purely absorbing medium and then expanded his research to include thermal multiplying media using coupled simplified one-dimensional neutron diffusion and burn-up equations [8]. Van Dam [9] expanded on Seifritz's work in absorbing media and extended it to diffusive media by considering moderator in absorbing media. Later, Van Dam [10] investigated nuclear burn-up in a thermal medium by considering simplified analytical model reactivity feedback. Van Dam [11] did further study of burn-up wave (used terminology “self-stabilizing criticality wave”) for a cylindrical thermal pebble-bed type high-temperature gas cool reactor (HTGR). Over several years, a substantial amount of analytical and numerical research has been conducted on the B&B concept for thermal, epithermal, and fast neutron energy regimes [12–36]. In recent years, Terra Power, a private company from the U.S. and funded by Mr. Bill Gates, has started working on the new concepts of traveling wave reactors using fast reactor technology [37–39]. Radial fuel reshuffling methodology has been utilized to maintain power density uniformly distributed inside the core and within safety limits [40,41].

Various private and public organizations are currently working on the detailed physics and engineering designs of B&B-based reactors using complex neutronic codes (Monte Carlo N-Particle Transport Code, Serpent, Eranos, etc.) and multigroup cross section data files. Availability of these codes is not only scarce but also computationally time intensive. As most of the researchers are currently focusing on the concepts and preliminary designs [42–46], it is meaningful to assess burn-up waves for simplified geometry and perform analysis on the wave characteristics. This qualitative and quantitative analysis can be beneficial for the practical design and development of this kind of reactor.

The Indian Institute of Technology Kanpur (IITK) has been actively engaged in the study related to the development and characterization of burn-up wave in absorbing, diffusive, and multiplying medium (both thermal and fast) [31–34]. A brief overview of the significant findings made by IITK in this field is presented below to emphasize the current state of the art, identify existing limitations, and establish the solid objective of the work presented in this paper in continuation to previous work.

The possibility of nuclear fuel burn-up wave in the epithermal neutron energy regime was demonstrated by Ray et al. [35] by performing numerical simulations. Single energy group neutron diffusion equations with the fairly simplified burn-up model in semi-infinite slab geometry have been used for calculation in earlier research in IITK [31,32]. Kumar and Singh [33] improved the computational model by modifying it for the cylindrical geometry in fast multiplying medium, and transverse neutron leakage was represented by using radial buckling approximation. The major limitation of the radial buckling concept is its inability to properly describe the radial inhomogeneity caused by the nonuniform burning of fuel nuclides in the radial direction. This sort of scenario emerges during the inclusion of radial neutron reflector in reactor architecture. Hence, the radial buckling concept can provide effective qualitative resemblance compared to the two-dimensional model, although quantitative discrepancies can be identified as reported by Fomin et al. [19]. Ray et al. [36] improved the limitations due to the simplifications in the research of Kumar and Singh [33] by using a two-dimensional cylindrical model of a bare B&B reactor with multigroup neutron diffusion and detailed burn-up equations.

The objective of this study is to demonstrate the potential significance of the newly introduced characterization parameters (transient time (TT) and transient length (TL)) on the development of radial neutron reflector in B&B-based reactor design. These two parameters are utilized to characterize the transient aspect of the fuel burn-up wave. TT and TL are specified as the time and distance required to achieve the steady-state nature of the burn-up wave. Steady-state characteristics are defined by the velocity of wave propagation and the width of the reaction zone expressed as FWHM (full width half maximum) and FW10M (full width 10% of the maximum). In this study, coupled multi-energy group neutron diffusion equations with nuclide burn-up model in U–Pu fuel cycle are solved using the comsolmultiphysics software package. Two-dimensional cylindrical geometry with azimuthal symmetry is considered for calculation. Heterogeneity in material compositions in the radial direction is introduced to illustrate the influence of radial neutron reflector on the evolution and propagation of nuclear fuel burn-up wave. Sensitivity study of transient and steady-state parameters of fuel burn-up wave with respect to the thickness of radial reflector is also presented.

Mathematical representation of the fast reactor in a multiplying medium for the U–Pu fuel cycle is depicted in Fig. 1. A finite sized cylindrical reactor with azimuthal symmetry is considered for calculation. Radius of the core and the thickness of the reflector are represented by R and R_ref, respectively. Two homogeneous zones are taken into account in the reactor core. First zone is the ignition zone (on the reactor's bottom side), and it is filled with natural uranium (99.28% of ²³⁸U and 0.72% of ²³⁵U) and Pu isotopes. Plutonium enrichment of 10% is chosen in the ignition zone. Second zone (breeding zone) is adjacent to the ignition zone, and it contains natural uranium as fuel. Structural material (steel) and coolant (Na) are available in both zones. Volume fractions of fuel, coolant, and structural material in each zone are chosen as 0.42: 0.33: 0.25, respectively. Radial reflector considered for the calculation consists of steel and Na. Material fractions and isotopic compositions of different zones of the reactor are presented in Table 1. These values of volume fraction are fairly close to the material composition of the actual fast breeder reactor (FBR) [47–49]. External neutron source of $jex$ is applied at the bottom of the reactor to initiate the burnup process.

Source term (⁠$χg∑g′=1G(νΣf)g′ϕg′$⁠) is not included for calculating neutron diffusion equations in nonmultiplying neutron reflector region (no presence of nuclear fuel material). In the above equations, radial and axial direction is represented by $r,z$⁠, respectively; j in $Σj$ stands for ²³⁸U, ²³⁵U, ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu, and ²⁴²Pu material; $ϑg$ and $Dg$ denote neutron velocity of a particular energy group and group diffusion coefficient of the medium, respectively; $Σa$ and$Σf$⁠, respectively, represent macroscopic absorption and fission cross section; $Σs,g′→g$ and $Σs,g→g′$ represent macroscopic scattering cross section (group $g′$ to$g$ and group $g$ to$g′)$⁠, respectively; σ_a, σ_c, and σ_f represent microscopic absorption, capture, and fission cross section, respectively; $ν$is the number of neutrons emitted per fission; $ϕg$ represents group scalar neutron flux; N₈, N₅, N_Np, N_Pu9, N_Pu0, N_Pu1, and N_Pu2 represent nuclide density of ²³⁸U, ²³⁵U, ²³⁹Np, ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu, and ²⁴²Pu nuclides. Fission product pair created at every fission process is considered to be one nuclide, and fission product density is denoted by the symbol$NFP$⁠. $τβ$ represents the mean life-time for the $β$-decay process of ²³⁹Np (≈ (2.35 days/ln2 = 3.4 days)), and $λ$ represents the decay constant of the nuclide.

The term $qg(r,z,t)$denotes external neutron source for group g in the neutron diffusion Eq. (1). Neutron source is utilized for the initiation of the burn-up wave. External neutron sources can be thought of in two ways: one is neutron creation in the ignition zone, and the other is neutron current impinges on the reactor's bottom boundary at z = 0. So, the necessary boundary conditions presented in Fig. 1 for the cylindrical reactor geometry are as follows:

• Zero neutron current is entering through the top side of the geometry

  $(ϕg(r,z)+2Dg∂ϕg(r,z)/∂z)z=zt=0$

  (10)
• An external neutron current of $jex$ is applied at the bottom boundary to initiate the burn-up wave

  $(ϕg(r,z)−2Dg∂ϕg(r,z)/∂z)z=zb=2jex$

  (11)
• Zero flux boundary condition is imposed at the external surface of the radial reflector

  $ϕg(R,z)=0$

  (12)

Eight neutron energy groups of microscopic cross section data for sodium FBR have been taken from Walter et al. [48]. These eight groups of cross sections were constructed by compressing a 42 group cross section dataset derived from the ENDF/B nuclear data library using the shielding-factor method. Since the present investigation is more concerned with preliminary analysis than with actual design, these cross sections can be utilized to solve the neutron diffusion and burn-up equations.

Stationary calculation of the neutron diffusion equation is required to find the initial critical reactor configuration. Normalization of scalar neutron flux for the critical fast reactor is performed by considering average power density to a value of about$10−5 W/cm3$[15,18]. Necessary conditions for stationary calculation are $∂ϕg/∂t=0$ and $q(r,z,t)=0,$while in case of a nonstationary problem $keff$ is considered to be 1. The ²⁴¹Am and ²⁴³Am isotopes are excluded from the burn-up model as the decline in their concentrations induced by absorption reactions is minimal in contrast to the processes considered [48]. ²³⁹U is disregarded from the burn-up model due to its small half-life (⁠$∼23.5 min)$⁠. The main contribution of ²³⁹Np in the burn-up calculation is considered due to its beta decay and conversion to ²³⁹Pu nuclide. For simplicity, the loss term due to capture is not included in the ²³⁹Np evolution model. The annihilation of fission products caused by neutron absorption is not taken into account. However, the absorption of neutrons in fission products is considered in the model. These assumptions do not compromise the principle of investigations on finding the influence of radial neutron reflector on the B&B-based reactor using a parametric approach but to simplify it to tackle intensive computation.

Coupled nonlinear partial differential equations can be derived by using any standard numerical techniques such as finite difference method (FDM), finite element method (FEM), or Monte Carlo method. In this study, commercial multiphysics software package comsol has been employed for both stationary and transient calculations. This software package appears to be a promising alternative for neutronic computation owing to its capability to incorporate diverse physics. comsol utilized a finite element analysis technique to solve the partial differential equations in integral (weak) form. In this technique, unknown dependent variables are discretized as a summation of a set of basis or shape functions defined on elements.

Table 2 shows the formulation of diffusion equations expressed in terms of generalized Eq. (13).

For stationary calculation, values of the coefficient in generalized Eq. (14) are the same as mentioned in Table 2. Table 3 displays the representation of individual 16 equations (neutron diffusion and burn-up evolution models) described in terms of generalized Eq. (14) for the case of transient calculation. Top side, bottom side, and external surface boundary conditions of the geometry are described by Dirichlet boundary conditions.

Implicit backward differentiation formula has been employed as the time-stepping technique for transient computation in comsol. The solver approximates the time-step necessary for error convergence and damping factor setting. The maximum time-step is controlled by the solver's intermediate stepping option.

Table 4 shows the cross section values of two-energy groups and geometrical data of the rectangular geometry (⁠$L1×L2)$⁠.

The obtained value of $keff$ from comsol code is 1.146848, whereas the reported value in Ref. [50] is 1.14685. Difference in the result between the two analyses is approximately 0.2 pcm. The calculated results are consistent with the reference value, thus, justify the capability of comsol code in solving effective multiplication factor.

Mesh independence study has been performed for steady-state analysis of the two-dimensional cylindrical reactor geometry. The reactor core length of 500 cm, a radius of 90 cm, and a reflector thickness of 30 cm have been considered for the study. Length of the ignition zone, which is 75 cm, is determined by the stationary criticality calculation and to maintain the initial reactor close to critical (⁠$keff≈1)$⁠. Calculations have been carried out with different mesh element sizes. For the mapping of three domains of two-dimensional geometry (ignition, breeding, and reflector), structured quadrilateral mesh has been adopted. Each domain is partitioned into a number of meshes with the maximum mesh element size constraint imposed by comsol solver. Maximum amplitude of the summed group scalar neutron flux (⁠$∑g=18∅g(0,z)$⁠) along the axial direction (at radius r = 0) has been chosen as the target parameter for the mesh independence study. Details of the mesh independence study are presented in Table 5. Finally, A3 option has been considered for calculation purpose.

Numerical simulation has been conducted on the two-dimensional cylindrical reactor geometry without a reflector to evaluate the impact of neutron reflector in B&B-based reactor. External source with an intensity of 3$×1012 cm−2 s−1$ has been applied at the bottom boundary for the initiation of the burn-up process. Figure 2 shows that burn-up wave fails to propagate along the media for this particular reactor arrangement. Evolution of burn-up wave in the axial direction for the bare reactor configuration is depicted in Fig. 3. The symbol$∅t$is basically the summed eight groups of scalar neutron flux$, ∑g=18∅g(r,z)$⁠.

Same reactor design utilized in the previous computation is subjected to a second analysis with a reflector of 30 cm thickness. Results confirm the evolution and propagation of fuel burn-up wave across the media, and it can be observed in Fig. 4.

Axial profile of summed group scalar neutron flux and power density $P(0,z)$ at radius r = 0 is displayed at mentioned time intervals in Figs. 5 and 6, respectively.

At the beginning of reactor operation, the neutron flux experiences a rapid increase due to the prompt emergence of the burn-up wavefront, reaching its maximum value around 140 days (see Fig. 5). For better visibility of the overall burn-up wave, the neutron flux and power density values at 140 days are scaled down by multiplying them by 0.09, following the methodology used by Fomin et al. [18,20]. After this initial surge, there is a decrease in the neutron flux over the next 400 days, leading to the initiation of a stable fuel burn-up wave formation along the reactor length. Build-up and propagation of burn-up wave is clearly depicted in Fig. 7, which shows the envelope of the maximum amplitude of summed group scalar neutron flux with time. During the steady-state phase, burn-up wave propagates almost uniformly between about 1050 and 2600 days. After this period, the wave begins to attenuate toward the top side of the reactor boundary, eventually reaching extinction around 3500 days. Calculated neutron flux and power density profile are in the same order as the values reported in Ref. [18].

Radial distribution of summed group scalar neutron flux for the reactor with and without radial neutron reflector in the breeding zone (at$z=110 cm$⁠) is depicted in Fig. 8. Flux profile at 1750 days for the bare reactor and 90 days for the reflected reactor are utilized as fuel burn-up wavefront starts to evolve around that time for both the cases (see Figs. 3 and 5). Increase in neutron flux in the reflector region is clearly visualized from Fig. 8. The reason for this is that with the presence of reflector, neutrons that would have escaped from the system reflect back into it, increasing the neutron population in that region.

For the reactor geometry with reflector, it can be noticed that spatial neutron flux initiates and builds up over a distance of 257.5 cm and at a time of 1050 days. The shape of the neutron flux is nearly preserved beyond this time and distance until the commencement of its extinction. Thus, TT and TL for this case are 1050 days and 257.5 cm, respectively. Figure 9 illustrates the value of FWHM and FW10M, TT and TL. FWHM and FW10M for this scenario are 103.8 cm and 210.5 cm, respectively, while wave velocity is found to be 32.49 cm/yr.

Figure 10 depicts the spatial profile of the density of ²³⁵U nuclide in the axial direction at radius r = 0. The concentration of ²³⁵U nuclide declines over time owing to its capture and engagement in the fission process. Similar characteristics can be observed for the ²³⁸U nuclide density as it transformed into ²³⁹Pu nuclide by two beta decay process (see Fig. 11). Due to the lower concentration of ²³⁸U nuclide in the ignition region, there is a noticeable rise in density along the boundary of the ignition and breeding zones.

Axial profile of ²³⁹Pu nuclide density with time advancement is depicted in Fig. 12. Initial change in ²³⁹Pu concentration is slow near the bottom reactor boundary, and gradually it starts to accumulate in the breeding zone near the top edge due to the transmutation of ²³⁸U nuclide. Steady-state nature of ²³⁹Pu density can be witnessed in the middle region of the reactor. Other plutonium isotope's (²⁴⁰Pu, ²⁴¹Pu, and ²⁴²Pu) concentration displays similar trends with time evolvement as ²³⁹Pu nuclide. Figure 13 shows the axial profile of fission product density. Fission products also maintain its uniform nature in the middle portion of reactor region. More than 50% fuel burn-up depth is achieved in the overall reactor volume (see Fig. 13) except near the boundary of the reactor.

Steady-state criticality calculation is performed for the two reactor configurations with and without reflector. It is found that for the same fuel composition in the reactor zone, radius of the bare reactor has to be increased to 107.24 cm to reach the same value of $keff$ that was achieved in case of the reactor with a radial reflector thickness of 30 cm. Rest of the geometrical value (ignition zone length and reactor length) remains same for both design as mentioned in the earlier part of this section. Therefore, the reflector savings for the reactor geometry with a reflector (thickness of 30 cm) is 17.24 cm. Figure 14 depicts the different values of reflector savings with the variation of reflector thickness. The figure shows that the variation in reflector savings is considerable at lower values of reflector thickness, whereas for the higher values, variation is minimal. Decrement of about 5.6% in reflector savings is observed for the decrease in reflector thickness from 50 cm to 28 cm.

Parametric study of steady-state and transient characteristics of the burn-up wave with respect to the thickness of radial neutron reflector is presented in Table 6. Radius of the reactor and ignition zone length remains 90 cm and 75 cm for all the cases, respectively. Steady-state criteria for the determination of characterization parameters of the fuel burn-up wave have been chosen at the region where the maximum reaction rate is horizontal. Slop of the maximum reaction rate is not uniform in some scenarios. In those circumstances, percentage change of maximum reaction rate between two day intervals is evaluated. Initiation of the steady-state is assumed when the percentage change is almost equal.

A list of observations can be made from the results obtained in Table 6.

1. From Fig. 15, it can be seen that TT decreases with the increase in the thickness of radial reflector. Variation is prompt for the lower values of reflector thickness (value below 40 cm), although it tends to get slowed down for the higher values of thickness. Decrement of about 42.1% in TT is observed for the increment of reflector thickness from 28 cm to 50 cm.

2. Figure 16 depicts the decrease of TL with the increment in radial reflector thickness. Variation of TL with respect to reflector thickness (decrease of about 1.7% in TL due to increase in reflector thickness from 28 cm to 50 cm) is quite sluggish compared to the variation observed in the case of TT.

3. Velocity of wave propagation increases with the increase in reflector thickness. Wave velocity increases quite steadily for the lower values of thickness, although it remains slow for the higher values (reflector thickness above 40 cm).

4. Variation of reflector thickness has a negligible influence on reaction zone width, which is represented by FWHM and FW10M.

5. Gradual increase in asymptotic value of summed group scalar neutron flux is observed with the radial reflector thickness.

Relevance of the characterization parameters in the design of radial reflector for B&B reactor can be discussed by analyzing Table 6 and Fig. 17 mentioned below.

Figure 17 illustrates that for the reflected reactor with radial reflector thickness of 28 cm, envelope of the maximum amplitude of summed group scalar neutron flux is able to maintain its shape for more than 3500 days. This feature ensures the reactor operation for that particular configuration for more than 3500 days with relatively lower values of neutron flux and wave velocity (refer to Table 6). The approximate duration of the burn-up wave from the start to the initiation of extinction for a particular configuration is denoted as $Ts$⁠. The percentage of stable reactor operation can be calculated using the following formula:

Percentage of stable reactor operation (%) = $[(Ts−TT)Ts×100]$

Figure 17 shows that for a reactor configuration with a radial reflector thickness of 30 cm, the neutron flux begins to attenuate around 2710 days until it reaches extinction. Thus, the value of 2710 days is taken as the value of $Ts$ for this configuration. Additionally, Table 6 indicates that TT for this configuration is 1050 days. Consequently, the percentage of stable reactor operation can be calculated as follows:

Percentage of stable reactor operation for the reactor with radial reflector thickness of 30 cm = $[((2710−1050)/2710)×100]≈$ 61.25%. This parameter for other reactor configurations has been calculated in a similar manner.

Increment in reflector thickness leads to reduction in both TT and TL values, but it introduces a considerable rise in wave velocity and neutron flux. This causes the fuel material in the breeding zone of the B&B reactor to burn out faster due to the availability of more number of neutrons, resulting in a shorter period of stable reactor operation, as indicated in Table 6. Lower duration of stable reactor operation is not practically desirable in terms of an economic point of view as the key objective of a B&B reactor is to operate for a longer period while preserving its shape. Additionally, the high value of neutron flux will be of major concern as it exceeds the overall heating removal capability of the currently existing coolants and reactor safety systems. Hence, special attention should be employed to the optimization of these characterization parameters to ensure a longer reactor operating life while maintaining it within the design safety limit.

In this present analysis, a simplified reactor startup strategy has been employed, and sensitivity evaluations of the characterization parameters have been conducted based on this approach. This consideration serves to justify the significance of the newly introduced parameters on the preliminary B&B reactor design. Nevertheless, during a real reactor design scenario, a smooth startup scheme has to be considered since an unanticipated initial surge in neutron flux and power can cause a considerable amount of damage to the reactor system (see Fig. 17). Fomin et al. [52] proposed one of the approaches to tackle this undesired phenomenon by evaluating appropriate fuel and absorber concentration in the reactor ignition zone.

For the reflector thickness of 25 cm in a specific reactor configuration, results show that fuel burn-up wave develops and shifts along the reactor axis for relatively longer period of time. However, burn-up wave dies out before it reaches the top end of the reactor (see Fig. 18). This investigation demonstrates that using the adjustment in reflector thickness reported by Fomin et al. [51], reactor power can be regulated as well as terminated without any alteration in the reactor core. The value of neutron flux of the developed burn-up wave as depicted in Fig. 5 is found to be in the range of $1016 cm−2 s−1$ in this study. This flux value is higher compared to the actual fast reactor design value, which is generally in the range of $1015 cm−2 s−1$ [53]. Although it can be observed in Ref. [51] that the neutron flux value of the burn-up wave is in the range of $1014 cm−2 s−1$⁠. In the calculation of Fomin et al. [51], different fractions of fuel, coolant, and structural material, as well as different material compositions for the reactor (such as Pb–Bi instead of Na as coolant), were incorporated, also the neutronic parameters from 26 neutron energy groups [53] were considered in the study. However, in the present analysis, emphasis is given to qualitative exploration rather than quantitative study since this is not a design-focused study.

Effects of radial neutron reflector on the characteristics of nuclear fuel burn-up wave have been investigated in a fast neutron multiplying medium for a two-dimensional cylindrical reactor with azimuthal symmetry. Multigroup neutron diffusion equations (eight energy groups of neutron) with a nuclide evolution model have been included in the computational model. Structural materials (steel) and coolant (Na) have been taken in both homogeneous zones (ignition and breeding) of the reactor and as the material of radial reflector. comsolmultiphysics software, based on finite element numerical technique, has been considered for simulation purpose. Initial compositions of the fuel components and geometrical values have been adjusted using criticality calculation to maintain the initial reactor to a near-critical state. Two parameters named TT and TL are utilized to represent transient characteristics of the fuel burn-up wave. Steady-state part of the wave is expressed in terms of wave velocity and reaction zone width (FWHM and FW10M).

The study has been carried out for the reactor configurations considered here when reflector is ignored in the design and subsequently when radial reflectors of various thicknesses are taken into account. The absence of a radial reflector prevents the development of a fuel burn-up wave for the reactor with a radius of 90 cm (reactor length of 500 cm and ignition zone length of 75 cm) in the present analysis. However, as the reflector (thickness of 30 cm) is employed, the wave development grows gradually and propagates along the reactor length. Fuel burn-up depth of more than 50% is achieved for that particular reflected reactor configuration. Reflector saving of 17.24 cm is discovered for that specific case. The analysis shows a decline of about 5.1% in the reflector savings owing to the decrease in reflector thickness from 50 cm to 28 cm.

Further sensitivity study has been conducted on the characterization parameters of fuel burn-up wave with the variation of reflector thickness while maintaining geometrical configuration of the reactor same for all cases. Studies show that TT decreases by about 42.1% and TL by about 1.7% with the increment of radial reflector thickness from 28 cm to 50 cm. The variation is prompt for the lower values of reflector thickness, whereas, with the increase in thickness beyond 40 cm, changes slow down. Velocity of wave propagation and neutron flux increase with the increment in reflector thickness. Impact of radial reflector thickness on the reaction zone width (FWHM and FW10M) is fairly nonexistent. It is found that for a reflector of thickness 25 cm, fuel burn-up wave fails to propagate the whole reactor length.

Reactor configuration with a reflector thickness of 28 cm is determined to be capable of operating for more than 3500 days with a wave velocity of 20.63 cm/yr. However, unsatisfactory TT (1259 days) and TL (258.8 cm) values necessitate the optimization of characterization parameters by analyzing the qualitative and quantitative outcomes of its sensitivity study with radial reflector thickness. This research could certainly assist in the practical design of the B&B reactor.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$Dg$ =

group diffusion coefficient of the medium, m

$Jex$ =

neutron current density entering the bottom boundary of the cylindrical geometry, neutrons/$m2$s

$l$ =

diffusion length, m

$N$ =

neutron density, 1/m³

$N5$ =

nuclide density of 235 U, 1/m³

$N8$ =

nuclide density of 238 U, 1/m³

$NFP$ =

nuclide density of fission products, 1/m³

$NNp$ =

nuclide density of 239 Np, 1/m³

$NPu0$ =

nuclide density of 240Pu, 1/m³

$NPu1$ =

nuclide density of 241Pu, 1/m³

$NPu2$ =

nuclide density of 242Pu, 1/m³

$NPu9$ =

nuclide density of 239Pu, 1/m³

²³⁹Np =

neptunium-239, symbol

²³⁹Pu =

plutonium-239, symbol

²⁴⁰Pu =

plutonium-240, symbol

²⁴¹Pu =

plutonium-241, symbol

²⁴²Pu =

plutonium-242, symbol

$qg$ =

external source of neutrons for group g, 1/$m2$s

r =

distance in radial direction, m

t =

time, s

²³⁵U =

uranium-235 symbol

²³⁸U =

uranium-238 symbol

$ϑg$ =

neutron velocity of a particular energy group of the medium, m/s

z =

distance in axial direction z, m

Greek Symbols

$λ$ =

decay constant of nuclide, 1/s

$λPu0$ =

decay constant of 240Pu, 1/s

$λPu1$ =

decay constant of 241Pu, 1/s

$λPu2$ =

decay constant of 242Pu, 1/s

$λPu9$ =

decay constant of 239Pu, 1/s

$λ5$ =

decay constant of 235 U, 1/s

$λ8$ =

decay constant of 238 U, 1/s

$Σa$ =

macroscopic absorption cross section, barn

$Σf =$ =

macroscopic fission cross section, barn

$Σs,g′→g$ =

macroscopic scattering cross section (group $g′$ to$g$⁠), barn

$Σs,g→g′$ =

macroscopic scattering cross section (group $g$ to$g′)$⁠, barn

$σa$ =

microscopic absorption cross section, barn

$σa5$ =

microscopic absorption cross section of 235 U, barn

$σa8$ =

microscopic absorption cross section of 238 U, barn

$σag$ =

microscopic absorption cross section of a particular energy group of the medium, barn

$σc$ =

microscopic capture cross section, barn

$σc5$ =

microscopic capture cross section of 235 U, barn

$σc8$ =

microscopic capture cross section of 238 U, barn

$σcPu0$ =

microscopic capture cross section of 240Pu, barn

$σcPu1$ =

microscopic capture cross section of 241Pu, barn

$σcPu2$ =

microscopic capture cross section of 242Pu, barn

$σcPu9$ =

microscopic capture cross section of 239Pu, barn

$σcg$ =

microscopic capture cross section of a particular energy group of the medium, barn

$σf$ =

microscopic fission cross section, barn

$σf5$ =

microscopic fission cross section of 235 U, barn

$σfPu0$ =

microscopic fission cross section of 240Pu, barn

$σfPu1$ =

microscopic fission cross section of 241Pu, barn

$σfPu2$ =

microscopic fission cross section of 242Pu, barn

$σfPu9$ =

microscopic fission cross section of 239Pu, barn

$σfg$ =

microscopic fission cross section of a particular energy group of the medium, barn

$τβ$ =

mean life-time for the $β$-decay process of 239 Np, s

$ϕg$ =

group scalar neutron flux, 1/$m2$s

Acronyms and Abbreviations

B&B =

breed and burn

CANDLE =

constant axial shape for neutron flux, nuclear densities, and power during life of energy producing reactors

FBR =

fast breeder reactor

FDM =

finite difference method

FEM =

finite element method

FWHM =

full width at half maximum

FW10M =

full width at 10% of maximum

HTGR =

high-temperature gas cooled reactor

PDE =

partial differential equation

TT =

transient time

TL =

transient length

TMI =

Three-Mile Island

TWR =

traveling wave reactor