Corresponding author: Vladimir I. Korolev (vlikor2007@yandex.ru)

Academic editor: Yury Kazansky

At the present time, JSC Baltiskiy zavod has built and transported to the deployment site at Pevek

Reactor cores for nuclear powered floating facilities were developed in two major stages using different design approaches dictated by the demands of the time. Fig.

Diagram of the key requirements to the reactor cores of nuclear powered floating facilities and internal relationships between them for ships built in 1975–2007 (projects 1052, 10521, 10580) and for modern vessels (projects 20870 PEB, 22220 UAL).

The objective at stage 1 was to build a high-power reactor core, primarily for the needs of the Soviet Navy. It took quite a long time to achieve the required core performance since no desired reliability could be ensured because of the core’s high power rating of ~ 165 MW/ m^{3} (KLT-40). It should be noted that the heat density of fixed reactor cores was ~ 80 MW/m^{3} (VVER-440) and ~ 110 MW/m^{3} (VVER-1000). Long-term integrity of the fuel cladding was also an issue hard to resolve. Another problem caused by the high power rating of ship reactor cores was surface boiling of the coolant on the fuel cladding at power values of over 60% of the rated power. This phenomenon was attributed to the "nodular (spot) corrosion” of zirconium fuel cladding (

The conceptual approach was changed at stage 2 since cores were largely designed for civilian floating facilities, and it became possible to promote them at international market (the path towards globalizing Russian nuclear technologies proclaimed by Rosatom State Corporation). Further, the core developers and manufacturers were aimed at giving cores a better reliability and higher energy content. At the same time, it was not stringently required to make cores compact. More importantly, the challenges of the time set the goal to ensure that Russian transport reactor cores be recognized internationally. Later cores were therefore significantly improved. The monoblock (integral) reactor design (project 22220) made it possible to increase to a certain extent the core dimensions and to improve their energy content. Given the long-term perfection of the cores from evolution stage 1, one should expect the situation to be the same for the second stage core. The author proposes a procedure for the rapid preliminary modeling of reactor cores differing in design characteristics and performance.

Improving the energy resource and energy content of transport reactor cores requires a study to be undertaken into the core design and performance parameters. The integrity of fuel cladding depends to a certain extent on the maximum in-core heat flux density _{s}^{max} and the maximum specific accumulation of fission products in dispersion fuel _{fp}^{max}. The combination of the above parameters may have an independent value. For instance, the relation ξ = _{s}^{max} / _{fp}^{max} [MW·cm^{3}×m^{–2}×g^{–1}] characterizes the core heat density, on the one hand, and the energy content in terms of the loaded fuel amount and the energy resource in terms of the fuel cladding reliability, on the other hand. A higher value of this relation can be achieved at the expense of increasing the core power rating and reducing the fuel cladding reliability due to a greater probability of the core integrity loss or the core volume increase with the achieved burn-up depth. Formally then, for the emerging trends in the core design and construction in future the value ξ will tend to decrease: for the cores of the nuclear icebreakers in operation (evolution stage 1), ξ = 1.9 to 2.0; for the floating power unit (FPU) core (

The second core evolution stage also involves a number of trends towards (

reducing the core specific power rating, MW/m 3;

reducing the core specific heat flux, MW/m 2;

increasing the fuel heat-exchange surface area, m 2;

increasing the core volume, m 3;

increasing the core effective life, h;

increasing the specific fission product accumulation, g/cm 3.

A dimensionless combined parameter was proposed in (

_{ef} /_{fe}, (1)

where <

For thermal neutron reactors, the probability of the U-235 thermal neutron capture without fission is ~ 0.15, that is, each five or six fissioned U-235 nuclei give birth to the nucleus of the non-fissionable U-236 isotope the formation of which means that the fissionable nucleus is lost without energy generation. The probability of the Pu-239 transformation to a non-fissionable Pu-240 isotope is 0.26. Therefore, the presence of radiation capture reactions, which compete with the fission reaction, inevitably leads to an inefficient increase in the specific consumption of fissionable nuclides. As follows from the above, <

For illustration, Fig.

A semi-graphical model of nuclear fuel burn-up as a function of the core energy generation and operating conditions: _{e} – equilibrium mass of initial fuel and bred fuel; point 1 – in-core accumulation of fission fragments during reactor rated power operation and with the core’s normal thermophysical parameters; point 3 – accumulation of fission fragments during the reactor core operation at a power less than the rated power and with variation of the core’s thermophysical parameters – 0

With a high nuclear fuel burn-up, the concentrations of uranium-235 and plutonium-239 account nearly equally for the energy generation in the reactor (_{p}, is determined in the intersection of the

The highest accumulation rate of fission fragments takes place during the reactor rated power operation (0_{eff}).

The reactor is kept in a critical condition up to point 1 where the qualitative variation curve of the fissionable nuclide total mass (_{p,0} (estimated core energy content).

If, for example, the average coolant temperature is reduced, the total energy generation can be increased to the value _{p}^{+}. The reactor is kept critical up to point 3 (see Fig.

For transport reactor cores, the average specific consumption of U-235 for the core life is determined as <^{*} = _{5}^{0}/_{p,0}, g/(MW×day) (<^{*} = 1.86 – 2.1) where _{5}^{0} is the initial core U-235 weight load.

The dimensionless combined parameter for transport reactor cores can be also found from the relation

_{cl}>]^{2}(1 – _{sc}^{2}), (2)

where _{sc} =<_{sc}>/(1 – 2<δ_{cl}>) is the correction factor taking into account the presence of the swelling compensator (SC) in the dispersion fuel composition; δ_{cl} is the fuel cladding thickness; <δ_{cl}> = δ_{cl}/_{fe} is the relative fuel element cladding thickness; <_{sc}> = _{sc}/_{fe} is the relative equivalent diameter of the swelling compensator; and _{fe} is the fuel element diameter.

The value of the parameter

Using the dimensionless parameter

For a circular fuel element with the cladding diameter _{cl}^{fe} = 5.8 mm and the cladding thickness δ_{cl} = 0.5 mm, the parameter _{cl}^{fe}= 7 mm, δ_{cl} = 0.5 mm, then _{cl}^{fe} = 5.8 mm and δ_{cl} = 0.5 mm, and the swelling compensator diameter being _{sc} = 3.7 mm, the value _{cl}^{fe} = 9.1 mm, δ_{cl} = 0.65 mm, the gap width being δ_{gap} = 0.125 mm, and the internal hole diameter being _{hole} = 1.4 mm, the value

On the whole, in transport reactor cores, the value of the parameter

The higher is the parameter

An analysis of advanced fuel cladding materials has been undertaken at the National Nuclear Laboratory in Great Britain. Five major types of fuel cladding materials were considered with a different level of availability for being commercialized: improved zirconium alloys; ferrite and martensite steels; nickel-, vanadium- and chromium-based heat-resistant alloys; niobium-, tantalum-, molybdenum- and tungsten-based high-melting alloys; ceramic-based materials including fiber composites based on silicon carbide; MAX-ceramics, and zirconium carbide (

Expressions (1) and (2) can be used to determine the required diameter of the fuel cladding in meters (

where A = 4<_{eff} /(1-k_{sc}^{2}) is the integrated indicator, m.

Rapid modeling of transport reactor cores requires a connection to be established between the core design characteristics (circumscribed diameter and height), nuclear fuel load, number of fuel elements, etc. with the performance indicators, including maximum heat flux density, maximum specific accumulation of fission products (characteristics relation), effective life, and specific nature of the fuel design. The above performance indicators form a part of the integrated indicator

In accordance with (

where _{p,0} is the reactor facility rated power, MW; _{d}^{*} = [_{s}k_{σ}(1+ω)/(π^{1/3} = [_{cell}_{σ}/(π^{1/3} is the coefficient allowing for the core design peculiarities: _{s}_{s}_{σ} = 1 – 4<δ_{cl}> + 4<δ_{cl}>^{2} – <_{sc}>^{2} is the share of the cross-section occupied by the fuel composition (_{σ} = 0.5 – 0.6); _{v}^{q}

If we assume that δ_{cl} = 0.5×10^{–3} m, then, with regard for (3), expression (4) can be transformed as

where ^{Σ}_{fe} =_{p,0}_{v}^{q}_{s}^{max} is the total area fuel heat-exchange surface.

The initial core U-235 weight load, with regard for expression (3), where δ_{cl} = 0.5×10^{–3} m, can be presented as

where γ_{5}^{fm} = (_{5}/_{fr})_{5}γ_{fr}_{fr} is the specific mass of U-235 in the fuel matrix volume unit, kg/m^{3}; _{5}, _{fr} are the molecular weight of respectively U-235 and the fuel rods; _{fr} is the volume fraction of the fuel rods in the dispersion fuel composition; _{5} is the fuel uranium-235 enrichment; and γ_{fr} is the specific weight of the fuel rods (uranium dioxide or intermetallic uranium).

The cores of ship reactors from evolution stage 1 (_{4}Si)_{3}, dispersed into the Silumin matrix. The molecular mass of intermetallic uranium is

_{imu} = _{5}×^{5}_{imu} + (1 – _{5})×^{8}_{imu} =

_{5}×643 + (1 – _{5})×646. (7)

It has been proposed that new cores should use uranium dioxide for fuel elements. Its molecular mass can be determined from the expression

_{ud} = _{5}×^{5}_{ud} + (1 – _{5})×^{8}_{ud} = _{5}×267 + (1 – _{5})×270. (8)

In accordance with the IAEA nuclear weapon nonproliferation requirement, the U-235 enrichment of the nuclear fuel used in reactor cores is limited to _{5} = 0.2. This requirement is applied to the cores from evolution stage 2 (_{2}) makes it possible to increase the U-235 load, all other things being equal, by a factor of 2.4 as compared with intermetallic fuel.

One cylindrical rod-type fuel element may contain a certain mass of fuel which is estimated using the expression

With regard for expressions (6) and (9), the number of fuel elements needed to be loaded into the core is estimated using the following formula

where _{core} is the core height.

The number of fuel elements in the core depends on the core diameter, the number of nodes in the core, the number and specific arrangement of the elements its comprises (fuel rods, FAs, burnable absorber rods, CPS absorber rods, WNS and SNS rods), and the ratio between the moderator cross-section area and the fuel composition area across the elementary cell. The number of FAs can be estimated using the formula

_{FA} = _{core}^{2}/(_{cell}_{σ}<^{fill}_{fe}>_{fe}^{2}_{nod}^{FA}), (11)

where _{nod}^{FA} is the number of nodes in the FA (chosen from the discrete series 61, 91, 127, 169, ...); and <^{fill}_{fe}> is the average coefficient of the FA filling with fuel elements (for stage 1 cores, ^{fill}_{fe} = 0.84 – 0.87, and for stage 2 cores, ^{fill}_{fe} = 0.81 – 0.83).

Fig. _{core}, _{5}, _{cell}, _{d}^{*}, _{σ}, _{fe}, and <^{fill}_{fr}>. The core modeling using the diagram requires allowances to be introduced for the variation of the above quantity values. With regard for the corrections, formulas (5), (6), and (10) are reduced to the form

Nomogram for the preliminary modeling of the key design and performance characteristics of the nuclear powered floating facility reactor cores

_{core} = _{core}^{(d)}_{n}^{d}^{*}, (12)

_{5}^{0} = _{n}^{im}_{n}^{z}k_{n}^{b}k_{n}^{k}^{σ}_{5}^{0(d)}, (13)

^{Σ}_{fe} = ^{Σ(d)}_{fe}/_{n}^{H}

One can additionally estimate the required fuel enrichment _{5}, the fuel matrix volume _{fm}, the number of the FAs loaded into the core, _{FA}, and the calculated core energy content _{p,0}:

_{5} = 0.3436×10^{–3}_{5}^{0} / (_{n}^{H} k_{n}^{b} k_{n}^{k}^{σ}_{n}^{im}_{fe}^{2}^{Σ}_{fe}), (15)

_{fm} = 0.518 _{n}^{H} k_{n}^{k}^{σ}_{fe}^{2}^{Σ}_{fe}, m^{3}, (16)

_{FA} = _{core}^{2} / (2.78 _{n}^{cell}_{n}^{k}^{σ} <_{n}^{fill}> _{fe}^{2}_{nod}^{FA}), (17)

_{p,0} = _{p,0}×_{sc}^{2})×10^{3} / (0.203ξ), MW×η, (18)

where _{core}^{(d)}, _{5}^{0(d)}, ^{Σ}_{fuel}^{(d)} are respectively the circumscribed core diameter, m; the initial U-235 load, kg; the number of fuel elements in the core; _{n}^{d}^{*} = (_{n}^{cell}_{n}^{k}^{σ}/_{n}^{m}_{n}^{im} = 0.418 is the allowance for intermetallic uranium; _{n}^{z}_{5}/0.141 is the allowance for the fuel U-235 enrichment; _{n}^{b}_{fr}/0.59 is the allowance for the volumetric content of fuel elements in the fuel composition; _{n}^{k}^{σ} = {1 – 4<δ_{cl}> + 4<δ_{cl}>^{2} – [_{sc}(1 – 2<δ_{cl}>)]^{2}}/0.55 is the allowance for the share of the fuel element cross-section occupied by the fuel composition; _{n}^{cell}=_{s}_{n}^{fill}> = <_{fe}^{fill}>/0.82 is the allowance for the average coefficient of the FA filling with fuel elements; and _{n}^{H}_{core}/1.2 is the allowance for the core height.

Eight assembly-type core design options, which can be installed in transport reactors with modular or integral designs, are considered as an example. Estimates are based on a chart for modeling the key characteristics of the nuclear powered floating facility reactor cores. The variables included integrated indicator _{fe}, core rated thermal power _{p,0}, maximum heat flux density _{s}^{max}, characteristic ratio ξ, estimated core energy content _{p,0}, total in-core fuel element heat-exchange surface ^{Σ}_{fe}, circumscribed core diameter _{core}, fuel U-235 enrichment _{5}, initial U-235 load ^{0}_{5}, and number of fuel elements in the core ^{Σ}_{core}.

The fuel element design characteristics were assumed to be in the following limits: relative cladding thickness <δ_{cl}> = 0.056 – 0.073; relative equivalent diameter of the swelling compensator <_{sc}> = 0.34 – 0.418; share of the fuel element cross-section occupied by the fuel composition, _{σ} = 0.55 – 0.61; volume fraction of the fuel elements in the fuel composition _{fr} = 0.59 – 0.678. Options 1 through 7 use fuel elements of uranium dioxide, and option 8 uses fuel elements of intermetallic uranium. The estimation results are given in Table

Estimation results for transport reactor cores.

Description | Option | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Integrated indicator A, mm | 4.9 | 4.9 | 6 | 7 | 6 | 4.9 | 4.9 | 4.9 |

Core rated thermal power Q_{p,0}, MW |
150 | 175 | 175 | 175 | 315 | 315 | 315 | 175 |

Maximum heat flux density q_{s}^{max}, MW/m^{2} |
1.17 | 0.63 | 0.74 | 0.83 | 1.31 | 1.13 | 1.0 | 0.63 |

Bulk power peaking factor k_{v}^{q} |
1.75 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 |

Characteristic ratio ξ, MW×cm^{3}/(m^{2}×g) |
1.3 | 0.79 | 0.79 | 0.79 | 1.3 | 1.06 | 1.06 | 0.96 |

Flattening factor m | 1.0 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 | 1.05 |

Estimated core energy content W_{p,0}, TW×h |
2.1 | 4.5 | 5.5 | 6.4 | 6.0 | 6.0 | 6.0 | 3.7 |

Total core fuel heat-exchange surface F^{Σ}_{fe}, m^{2} |
224 | 527 | 450 | 400 | 457 | 527 | 600 | 527 |

Circumscribed core diameter D_{core}, m |
1.19 | 1.56 | 1.56 | 1.56 | 1.56 | 1.56 | 1.62 | 1.56 |

Core height H_{core}, m |
1.2 | 1.64 | 1.64 | 1.64 | 1.64 | 1.64 | 1.7 | 1.64 |

Fuel U-235 enrichment z_{5} |
0.141 | 0.141 | 0.141 | 0.141 | 0.141 | 0.141 | 0.12 | 0.198 |

Core average specific power density <q_{v}^{core}>, MW/m^{3} |
112.4 | 57.3 | 57.3 | 57.3 | 100.7 | 100.7 | 90.1 | 57.3 |

Initial U-235 load G_{5}^{0}, kg |
167 | 498 | 498 | 498 | 500 | 495 | 479 | 291 |

Number of fuel elements in core n^{Σ}_{fe} |
8750 | 14970 | 10971 | 8778 | 12339 | 15164 | 16620 | 14970 |

Specific U-235 consumption <k^{*}>, g/MW×day |
1.9 | 2.6 | 2.17 | 1.87 | 2.0 | 1.98 | 1.9 | 1.9 |

Max. specific accumulation of fission products g_{fp}^{max}, g/cm^{3} |
0.9 | 0.8 | 0.94 | 1.05 | 1.0 | 1.07 | 0.94 | 0.65 |

Options 1 and 2 consider cores with equal integrated indicators _{v}^{fc}> and in the core <_{v}^{core}>, as well as the maximum heat flux density in the core _{s}^{max}. Besides, there is an increase in the estimated core energy content _{p,0}. The fuel element has been somewhat redesigned to address this problem. In particular, the equivalent diameter of the swelling compensator has been reduced <_{sc}>, which has led to a reduced adjustment factor taking into account the existence of the swelling compensator _{sc}, and an increased share of the fuel element cross-section occupied by the fuel composition _{σ}.

The volume fraction of the fuel elements in the dispersion fuel composition _{fr} has also been increased. With a roughly equal maximum specific accumulation of fission products _{fp}^{max}, the characteristic ratio ξ = _{s}^{max}/_{fp}^{max} for the second option will decrease due to the reduced maximum core heat flux density. Reducing the maximum heat flux density will require increasing the total core fuel heat-exchange surface ^{Σ}_{fe}. This problem can be addressed through increasing the number of the fuel elements ^{Σ}_{fe} and the core height _{core}. To install more fuel elements, the diameter _{core} needs to be increased while not varying the flattening factor value ^{0}_{5} and the fuel matrix volume _{TM} loaded into the core will be also larger. With the assumed fuel U-235 enrichment, there is a certain increase in the specific consumption of U-235 <^{*}> observed for the considered option.

The integrated indicators were successively increased for core options 3 and 4. At the same time, the core rated thermal power _{p,0}, the characteristic ratio ξ, the circumscribed diameter _{core}, and the key design concepts for the fuel elements (_{sc}, _{σ}, _{core}) and the fuel (_{fr}, _{5}) remained the same. An increase of _{fe} led to a reduced number of the fuel elements in the core and, therefore, to a smaller total fuel heat-exchange area ^{Σ}_{fe}. This circumstance caused a successive increase in the maximum heat flux density _{s}^{max} with no variation of the characteristic ratio corresponding to the growth in the maximum specific accumulation of fission products _{fp}^{max}. It should be also noted that these two parameters are associated, to a large extent, with the core reliability and, therefore, with a potential reduction in the specified core energy content (_{p,0} increasing simultaneously for the core option under consideration. The specific power densities in the fuel composition and in the core remain roughly the same for options 3 and 4. The specific consumption of U-235 however decreases successively.

The cores considered for options 5, 6, and 7 have an equally higher power, _{p,0} = 315 MW, and an estimated energy content of _{p,0} = 6×10^{6} MW×η. At the same time, options 5 and 6 have the combined indicator _{s}^{max} in the limits of 1.31 to 1.13, while _{core} = 1.56 m, the U-235 enrichment _{5} = 0.141, and the fuel element design parameters are constant. A smaller value of the combined indicator leads to a decreased maximum heat flux density while the maximum specific accumulation of fission products somewhat increases. A further reduction in the maximum specific accumulation of fission products can be achieved through _{core} somewhat increasing in the limits of 1.56 to 1.62 m (option 7). This provides for _{s}^{max} decreasing in the limits of 1.13 to 1.0 MW/m^{2} with _{fp}^{max} reduced simultaneously in the limits of 1.07 to 0.94 g/cm^{3}, that is, there is a decrease in the fuel element energy load and, potentially, an increase in the core energy content.

Intermetallic uranium is considered as the fuel composition for core option 8 with all of the core’s key parameters for option 2 remaining the same. This core has a smaller estimated energy content of _{p,0} = 3.7 TW·h with the allowable fuel enrichment limited to _{5} = 0.198.

An expression was obtained in (

^{2}(Δ_{core})^{1,6}/[((_{I} – 6.5)/0.18)^{2} –

0.25(Δ_{core})^{2}]}^{0.625}, (19)

where _{I} is the assumed operating pressure in the primary circuit; Δ_{core} is the core temperature difference; _{s}^{max})^{0.2}_{cl}^{fe}[(_{FC} – _{BA})_{v}^{q}_{core}]^{0.8}, (kW/m)^{0.2} is the parameter; _{FC} = _{cell}_{σ}/(1 + <_{gap}>) is the dimensionless coefficient taking into account the core design peculiarities and the existence of a water gap between the FA shrouds (for channel-type cores); <_{gap} > coefficient taking into account the relative fraction of the moderator in the interchannel space of the active zone; and _{BA} = 1/^{fill}_{fe} is the coefficient taking into account the installation of burnable absorber rods, the CPS absorber rods, displacers, etc. into the FA nodes with no fuel elements.

The parameter

Currently, channel-type cores have begun to be installed at nuclear powered floating facilities (the floating nuclear power unit, the

The best possible selection of the reactor core design and performance characteristics and the core perfection process will require studying a large number of options meeting the given conditions. The developed procedure for rapid modeling of reactor cores allows considering the required number of options with as small costs as possible. The accuracy of calculation using the proposed procedure makes it possible to estimate fairly correctly the key design and performance parameters of the core while excluding low-efficiency options for achieving the objective set. To this end, detailed calculations need to be undertaken in future with respect to the selected options using the available designer codes. The design approaches used for the RITM-200 reactor facility aim at a potential further improvement of reactor cores and permit increasing the volume (through the height increase), the enrichment (from 7 to 19.7 %), the energy content (from 4.5 to 7.0 TW∙h), and the service life (to 53000 h) (

This rapid modeling procedure can be also used for the preliminary design of water-cooled water-moderated reactor cores for small nuclear power plants.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2020, n. 4, pp. 50–62.