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The Ghana Research Reactor-1 (GHARR-1) core was modified with an addition of a 9.0 mm layer of beryllium to the top shim tray to compensate for reactivity loss due to fuel depletion after 19 years of operation. Neutronic and kinetic parameters have been predicted using Monte Carlo N-Particle Code version 5 (MCNP5) to determine whether they were within acceptable operating margins. Excess reactivity, control rod worth, moderator reactivity coefficient, delayed neutron fraction and neutron generation time have been predicted as 3.86, 6.98,
−0.1218 mk/
°C, 8.17507 × 10
^{−3}
Δk/k, and 8.147 × 10
^{−5} s respectively. These parameters compared favorably with those provided in the initial Safety Analysis Report.

The Ghana Research Reactor-1 (GHARR-1) has undergone core conversion from the use of 90.2% high enriched uranium (HEU) fuel to the use of 12.5% low enriched uranium (LEU) fuel. The conversion of the reactor core was done under a project supported by the International Atomic Energy Agency (IAEA) with the aim of reducing proliferation risks associated with HEU fuel [

During the 19 years of operation, the excess reactivity of the HEU core dropped from 4.0 mk to about 2.3 mk due to fuel depletion and accumulation of fission products, some of which are neutron poisons. To restore the excess reactivity to the required 4.0 mk, the HEU core was modified with an addition of 9.0 mm of beryllium to the top shim tray. This modification was supported by an assessment of various safety parameters [

The safety parameters that were updated in the final SAR for the HEU core are; thermal hydraulics, neutronics and kinetics respectively. Thermal hydraulic parameters are not discussed under this work but have been discussed in [

The reactor was designed by the China Institute of Atomic Energy (CIAE) and was commissioned in 1986. It was designed to operate with a nominal power of 30 kW and a maximum neutron flux of 1 × 10^{12} n/cm^{2}∙s. The core was designed to use HEU as fuel, beryllium as reflector, light water as a moderator as well as a coolant and natural convection for cooling. The reactor was used for educational purposes, human resource development in nuclear science and technology, neutron activation analysis, and production of short-lived radioisotopes for use in hospitals. The core had 354 lattice positions arranged in 10 concentric circles at a pitch distance of 10.95 mm. It had 344 uranium-aluminum (U-Al) alloy fuel elements, 6 dummy aluminium elements, 4 tie rods, 2 grid plates and a central guide tube for the cadmium control rod. It was surrounded by an annular beryllium reflector and rested on a block of beryllium reflector plate. The top part of the core had a shim tray which was used for adding layers of beryllium for increased neutron reflection and reactivity insertion into the core. The top shim tray design allowed for reactivity compensation due to fuel depletion and accumulation of fission products as the reactor operated.

The reactor was under moderated with a high negative moderator temperature reactivity coefficient (approximately −0.l mk/˚C for the temperature range

20˚C - 45˚C), and a low critical mass which rendered it inherently safe. The availability of these characteristics, assured safety of the reactor under an array of possible accident conditions, such as power excursion that would occur following an accidental insertion of reactivity into the core [

Neutron interactions with reactor materials cause different phenomena such as nuclear fission, induced radioactivity and associated decay heating. These interactions play a central role in creating the power distributions, within nuclear materials, that drive the heat transfer process in a reactor. They also help in the determination of reactor safety, core properties, reactivity control, reactor kinetics, xenon stability, fuel depletion, and are also important for isotope production [

An understanding of the time-dependent behavior of neutrons in a nuclear reactor is important in the analysis of nuclear reactor safety. The transient neutron flux behaviour resulting from a departure from criticality may arise during operational changes such as control rods movement, environmental changes such as those of boron concentration, or accidental disturbances in steady-state operating condition. This phenomenon, among others, may be classified as either reactor kinetics or reactor dynamics.

Reactor kinetics also known as reactor kinetics without feedback, is the study of the neutron flux’s time-dependence for postulated changes in the macroscopic cross sections. Reactor dynamics or reactor kinetics with feedback and with spatial effects, is the study of the time-dependence of a neutron flux on the macroscopic cross sections as a function of the neutron flux level [

Neutronics of a nuclear reactor is the study of the distribution and multiplication of neutrons in the reactor and its effect on reactor power distribution. Diffusion codes are important in the study of neutronics and also in the determination of the neutron flux from the solution of the neutron diffusion equation. The neutron flux is important in computing the power distribution [

The multiplication of neutronsis is important if a reactor has to sustain a chain reaction. In order to have a sustainable chain reaction, neutrons emitted during fission should be able to cause further fissions in other fissile or fissionable nuclei. The fission chain reaction is quantitatively expressed in terms of the effective neutron multiplication factor (K_{eff}). The neutron multiplication factor, expressed by Equation (1), is a ratio of the number of neutrons in the current generation and the number of neutrons in the preceding generation. This factor is dependent on various parameters that also vary with temperature and are linked to reactor safety.

k e f f = Number of neutrons in one generation Number of neutrons in the preceding generation (1)

When K_{eff} is equal to unit, the reactor is said to be critical, when above or below unit, the reactor is said to be super critical or subcritical respectively. The measure of a reactor’s departure from criticality is called reactivity (ρ) and is defined as a fractional change in neutron population per generation. Equation (2) shows the mathematical expression of reactivity.

ρ = k e f f − 1 k e f f (2)

The ability to predict correctly the effective multiplication factor (K_{eff}) and computation of reactivity (ρ) is fundamental to solutions of criticality problems. The K_{eff} and ρ are important in the computation of such parameters as; the delayed neutron fraction, control rod worth, shutdown margin, neutron life time, and neutron generation time (Λ).

Correct computation of the prompt neutron generation time for a nuclear reactor under design or whose configuration has been modified is important for prediction and optimization of various reactor parameters. The neutron generation time has a bearing on various interactions and environmental conditions within a reactor. The prompt neutron generation time (Λ) is the average time from birth of a neutron to a point it reproduces itself through fission. The prompt neutron generation time can also be defined as the inverse of the neutron production rate as expressed by Equation (3). This is applicable when considering a one-group diffusion-theory approximation.

Λ = 1 V v Σ f (3)

where V in the average neutron velocity, ν is the average number of neutrons produced per fission and ∑_{f} in the fission macroscopic cross section.

There is, however, a difference between neutron generation time and neutron life time. Neutron life time is the time period between the birth of a neutron and its absorption. The neutron life time can be used to determine the neutron generation time using Equation (3). In light water reactors, the prompt neutron generation time increases with fuel burn up. This may be caused by a reduction in fissile materials as well as absorption in the moderator which makes the prompt neutron live much longer [

One of the commonly used method for determining the neutron generation time is the 1/V absorber method. This method applies the K_{eff} and ρ to compute the neutron generation time. In this method, all reactor materials in the core model are doped with small amounts of a 1/V neutron absorber material. The 1/V neutron absorber material inserts a negative reactivity expressed by Equation (4).

ρ = − Δ k k = k u − k p t k u ∗ k p t (4)

where Κ_{pt} is the perturbed multiplication factor due to the 1/V neutron absorber material and Κ_{u} is the steady state unperturbed multiplication factor. According to Bretscher [_{p}), the atomic density (n) of the 1/V neutron absorber material (atoms/b-cm), the speed of thermal neutrons (V = 2.2E5 cm/s), and the absorption cross section (σ = 3837b) of the 1/V neutron absorber material. This relation is expressed by Equation (5).

− Δ k k = l p V n σ thus l p = − Δ k k V n σ (5)

The neutron generation time is given as the limit of l_{p} as the n approaches zero. Equation (6) gives the expression for the neutron generation time.

Λ = lim n → 0 l p = lim n → 0 − Δ k k V n σ (6)

Monte Carlo N-Particle Transport (MCNP) is a general-purpose, continuous-energy, generalized-geometry, time-dependent, coupled neutron/photon/ electron Monte Carlo Transport code. The MCNP code is useful for complex problems that cannot be modeled by computer codes that use deterministic methods. Various cases can be simulated in several transport modes: neutron only, photon only, electron only, combined neutron/photon transport where the photons are produced by neutron interactions, neutron/photon/electron, photon/electron, or electron/photon. The neutron energy regime is from 10^{−11} MeV to 20 MeV, and the photon and electron energy regimes are from 1 keV to 1000 MeV. The capability to calculate K_{eff} eigenvalues for fissile systems is also a standard feature. The individual probabilistic events that comprise a process are simulated sequentially. The probability distributions governing these events are statistically sampled to describe the total phenomenon. It consists of following each of the many particles from a source throughout its life to its death in some terminal category (absorption, escape, etc.). Probability distributions are randomly sampled using transport data to determine the outcome at each step of its life. In this work, MCNP was used because of its ability to model complex three-dimensional geometries and its extensive validation [

The Reactor Burn up System (REBUS) code has been used to generate an inventory of isotopes produced due to fission products and fuel burn up after 19 years of GHARR-1’s operation. The REBUS code was simulated under GHARR-1’s operation scheme of 2.5 hours per day, 4 days a week and 52 weeks a year. This was done to produce an inventory of isotopes that was reflective of GHARR-1’s operating period. This inventory of isotopes was used to update the material card’s fuel composition in GHARR-1’s MCNP5 model. The material card is used to define the composition of different materials in the reactor model. A 9.0 mm layer of beryllium was added to the top shim tray of the reactor core as can be seen in

Criticality problems in MCNP are simulated using the KCODE card which requires an input of the nominal number of source histories (N) per cycle, an initial guess of K_{eff}, the number of source cycles (I_{c}) to be skipped before K_{eff} can accumulate, the number of active cycles (I_{A}), and the total number of cycles (I_{t}) [

The control rod worth, delayed neutron fraction, neutron generation time, shutdown margin, and power peaking factor cases have been simulated with the following specifications in the model; N was set at 300,000, K_{eff} was given an initial guess of 1.004, I_{t} was set as 250 cycles with I_{c} set at 20 cycles and the remaining I_{A} set 230 cycles being active. These values were adequate to provide sufficient statistical accuracy in the results. The value for the number of cycles to be skipped was chosen to provide enough cycles for K_{eff} to statistically converge around a certain solution, allowing for the most accurate results to be obtained. The initial guess for K_{eff} of 1.004 was based on the K_{eff} for GHARR-1’s clean core value, thus this value was sufficient to allow for good statistical accuracy in the prediction of K_{eff}. A total of 69 million particles were used for each simulation.

Simulation for K_{eff} was performed with the control rod fully withdrawn and the KCODE card parameters set as described in Section 3.1. The K_{eff} obtained was used to compute the excess reactivity using Equation (7).

ρ = k e f f − 1 k e f f (7)

The core has been simulated with the control rod fully inserted into the reactor core. The K_{effin} obtained has been used to compute ρ_{sdm} using Equation (8).

ρ s d m = 1 − k e f f i n k e f f i n (8)

Two cases have been simulated to obtain the eigenvalues needed to compute the control rod worth. For the first case, the control rod was fully withdrawn, giving a multiplication factor K_{effout}, and in the second case, the control rod was fully inserted into the reactor core giving the multiplication factor K_{effin}. Equation (9) was used to compute the control rod worth (ρ_{crw}).

ρ c r w = k e f f o u t − k e f f i n k e f f o u t k e f f i n = ρ o u t − ρ i n (9)

Reactivity coefficients have been simulated with the following KCODE card input; N was set at 200,000, K_{eff} was given an initial guess of 1.004, I_{t} was set at 220 cycles with the first 20 cycles I_{c} being inactive and the 200 cycles I_{A} being active. A total of 40 million particle histories have been used.

To obtain multiplication factors required to determine the moderator temperature coefficient, changes were made to the TMP card and all Cell cards that contain water in the MCNP model for GHARR-1. The TMP data card allows for input of equilibrium energy (MeV) of atoms in the fuel at a particular temperature. The TMP card and all Cell cards containing water in the model were respectively updated with the equilibrium temperature (MeV) and density corresponding to the temperature (˚C) that was being simulated for. The multiplication factors were predicted for the temperature range 20˚C to 100˚C. _{eff} and their corresponding equilibrium temperatures, densities and percent voiding. Corresponding reactivities for the temperature ranges have been computed using Equation (8). Each moderator reactivity computed has been plotted against the corresponding temperature (˚C) to obtain the relationship between temperature (˚C) and reactivity. The reactivity response to change in temperature has been determined using the 2^{nd} order differential equation. Equation (10) has been used to compute the moderator temperature coefficients.

Temperature (˚C) | Equilibrium energy (MeV) | Density (g/cm^{3}) | Void (%) |
---|---|---|---|

20 | 2.5125e−08 | 0.99820 | 0.00 |

30 | 2.6124e−08 | 0.99567 | 0.18 |

40 | 2.6935e−08 | 0.99221 | 0.70 |

50 | 2.7847e−08 | 0.98807 | 1.02 |

60 | 2.8709e−08 | 0.98324 | 1.50 |

70 | 2.9571e−08 | 0.97777 | 2.00 |

80 | 3.0432e−08 | 0.97170 | 2.55 |

100 | 3.2156e−08 | 0.95835 | 3.91 |

α x = ∂ ρ ∂ x (10)

where α_{x} is the reactivity coefficient for parameter (x).

The MCNP5 code does not have a card to input percent voiding in water due to changes in temperature. An indirect method has been used where densities corresponding to percent void as presented in

All Cell cards containing water and the TMP card were respectively updated with densities and corresponding temperatures (˚C) for the range 20˚C to 100˚C. Respective multiplication factors have been predicted for each temperature. Corresponding reactivities for the temperature range 20˚C to 100˚C were computed using Equation (8). ^{th} order polynomial equation. Equation (10) was used to compute the moderator void coefficient.

The TMP data card was updated with temperatures for the range 30˚C to 600˚C. This range was adequate to cover the operating range for the U-Al alloy fuel whose melting temperature is 640˚C. It was also sufficient to give an indication of the variation of the fuel temperature coefficient under accident conditions. The multiplication factors were predicted for the temperature range 30˚C to 600˚C. Corresponding reactivities for the temperature range 30˚C to 600˚C were computed using Equation (8). The relationship between temperature and reactivity was determined using the 4^{th} order polynomial equation. Equation (10) was used to compute the fuel temperature coefficient.

The effective delayed neutron fraction (β_{eff}) has been determined using the Prompt method which requires prediction of the multiplication factor for two cases [_{p}) only without the delayed neutrons. The second case involves prediction of the multiplication factor (Κ_{eff}) which combines both prompt and delayed neutrons.

In this method, the TOTNU data card with entry NO has been used to obtain the multiplication factor for prompt neutrons only. The TOTNU data card with entry NO prevents the influence of delayed neutrons on multiplicity per fission. Thus, the multiplicity per fission is due to the average prompt neutron and subsequently Κ_{p} is predicted for all fissionable nuclides with available prompt values.

The TOTNU card without any entry has been used to predict the average number of neutrons from fission. The TOTNU card without any entry considers both prompt and delayed neutrons and the resultant effective multiplication factor expressed by Equation (1) is given as Κ_{eff}.

For both cases, the KCODE card parameters were set as described in Section 3.1. The values for Κ_{p} and the Κ_{eff} have been used to compute the delayed neutron fraction using Equation (11).

β e f f = 1 − k p k e f f (11)

The neutron generation time has been determined using the 1/V absorber method described in Section 2. Boron-10 has been used as a doping material because of its 1/V neutron absorption property. Eleven cases have been used to predict multiplication factors required for the determination of the neutron generation time (Λ). The first case was used to predict the unperturbed multiplication factor (Κ_{u}) an equivalent of the multiplication factor Κ_{eff} described by Equation (1) for a reactor without any doping materials added. The remaining 10 cases have been used to predict the perturbed multiplication factors (Κ_{pt}) for10 different concentrations of boron-10 in the range 6.0 × 10^{−8} - 15 × 10^{−8} atoms/b-cm. Concentrations of boron lower than 6 × 10^{−8} atoms/b-cm have not been used since they do not provide significant change in reactivity as it has been reported by Hanson A. L. and Diamond D. J. [_{p}) obtained using Equation (5) have been

Case | Boron-10 Concentration (atoms/b-cm) |
---|---|

1 | Normal Condition |

2 | 6.0 × 10^{−8} |

3 | 7.0 × 10^{−8} |

4 | 8.0 × 10^{−8} |

5 | 9.0 × 10^{−8} |

6 | 10.0 × 10^{−8} |

7 | 11.0 × 10^{−8} |

8 | 12.0 × 10^{−8} |

9 | 13.0 × 10^{−8} |

10 | 14.0 × 10^{−8} |

11 | 15.0 × 10^{−8} |

plotted against their corresponding boron concentrations. The neutron generation time (Λ) has been determined as the intercept on the Y-axis based on the graphical application of Equation (6).

_{eff} as required if the reactor is to operate normally. If the excess reactivity exceeds 1/2β_{eff}, the reactor would tend to prompt criticality with a power excursion within seconds rendering it impossible to control. The control rod worth was predicted to be slightly higher compared to the value reported in the SAR even though the two values were fairly in agreement. The higher value from this work could be attributed to the increased margin between reactivities ρ_{out} and ρ_{in} described by Equation (9). It is expected that the reactivity ρ_{in} inserted when the control rod is fully inserted into the reactor core would be lower for the core after 19 years of operation compared to the value provided in the initial SAR due to increased negative reactivity resulting from fuel depletion and accumulation of neutron poisons. This causes a wider margin between the two reactivities making the control rod worth slightly higher. The control rod worth is an important design parameter for the safety of a reactor.

The shutdown margin (ρ_{sdm}) was predicted to be slightly higher compared to the value provided in the initial SAR. This could be attributed to a reduced value of the multiplication factor Κ_{effin} for the control rod fully inserted into the reactor core. From Equation (8), the reduced Κ_{effin} implies an increased value of ρ_{sdm}.

Simulation Results | Irradiated Core | Sigma | Safety Analysis Report (SAR) |
---|---|---|---|

Control rod fully withdrawn | 1.00388 | 1.4 × 10^{−4} | 1.00400 |

Excess reactivity (mk) | 3.86 | - | 4.0 |

Control rod fully inserted | 0.99689 | 1.4 × 10^{−4} | - |

Control rod worth | 6.98 | 2.0 × 10^{−2} | 6.8 |

Shutdown margin | 3.12 | 8.0 × 10^{−2} | 3.0 |

Delayed neutron fraction (∆k/k) | 8.17507 × 10^{−3} | 4.0 × 10^{−4} | 8.08 × 10^{−3} |

Neutron generation time (∆k/k) | 8.147 × 10^{−5} | 2.0 × 10^{−6} | 8.12 × 10^{−5} |

Maximum power peaking factor | 1.3522 | - | 1.3525 |

Moderator reactivity coefficient (mk/˚C) | −0.1218 | - | −0.1 |

The reduction in the value of multiplication factor Κ_{effin} compared to that in the SAR is influenced by increased neutron absorption due to accumulated neutron poisons in the fuel after 19 years. Despite the addition of a layer of beryllium to the top shim tray, the neutron poisons accumulated in the fuel still add a small amount of negative reactivity into the reactor core. This negative reactivity causes the ρ_{sdm} to be slightly higher compared to that reported in the SAR.

Both power distributions for the core after 19 years and that of the core provided in the initial SAR tilt lower towards the top of the reactor core. This implies the power at the top of the core in both cases is lower compared to the bottom of the core. This could be attributed to high neutron leakage at the top of the core compared to the bottom parts of both cores. The top part of the initial core is bare and that of the core after 19 years is reflected by 9 mm of beryllium compared to 50 mm of beryllium for the bottom parts of both cores.

Both the core after 19 years and that provided in the initial SAR show a similar power density distribution. However, the distribution for the core after 19 years is lower compared to that of the core provided in the initial SAR for the bottom part of the power density distribution. The lowered power density distribution observed near the bottom for the core after 19 years compared to the

core provided in the initial SAR could be attributed to accumulated fission products including neutron poisons in the fuel. These neutron poisons are not present in the core provided in the initial SAR. Towards the top of the core, there is an overlap where the core after 19 years has a higher power density distribution compared to the core provided in the initial SAR. The power density distribution overlap observed could be attributed to increased neutron density due to the presence of a 9.0 mm of beryllium that was added to the top shim tray after 19 years of operation.

The maximum power peaking factor for the core after 19 years was predicted to be 1.3522 and that of the core provided in the initial SAR was predicted to be 1.3525. Both values are within the expected range or 1.3 - 1.5 for light water moderated research reactors.

_{p}) against boron-10 concentration. The neutron generation time (Λ) was determined as the intercept on the

neutron-lifetime axis. The mathematical computation is elaborated in Equations (3) to (6). The value for the neutron generation time for the core after 19 years of operation was determined to be slightly higher compared to that of the core provided in the initial SAR. This could be attribute, in part, to the reduced average number of neutrons produced per fission due to fuel depletion and accumulation of neutron poisons in the fuel despite the addition of 9.0 mm of beryllium to the top shim tray. In light water reactors, the prompt neutron generation time increases with fuel burnup. This may be caused by a reduction in fissile materials as well as absorption in the moderator which makes the prompt neutron live much longer. Mathematically, this could be explained using Equation (3) from which the variable is the number of neutrons produced per fission. Reduction of this value implies an increase in the prompt neutron generation time.

Moderator Void CoefficientThe moderator void coefficient for the temperature range 20˚C to 100˚C was predicted to be negative.

function of temperature for the reactor core after 19 years of operation.

α v ( T ) = 3.0 × 10 − 10 T 4 − 4 × 10 − 8 T 3 + 1.8 × 10 − 5 T 2 − 0.0016 T + 0.021643 (12)

It can be observed from

Neutronics and kinetic parameters of GHARR-1’s HEU reactor core after 19 years subject to addition of 9.0 mm of beryllium to the top shim tray have been observed to be within the acceptable operating margins. The parameters were in good agreement with those of the core provided in the initial SAR. The accumulation of fission products and neutron poisons in the fuel meat had some effects on various parameters despite addition of beryllium. This caused some parameters under this work to either be higher or lower compared to those provided in the initial SAR. However, neutronic and kinetic parameters were observed to be within the allowed margins and as such the reactor was safe to operate with the addition of 9.0 mm of beryllium which compensated for the reactivity loss. The results obtained under this work are fit to be used as a basis for comparing operating conditions of the LEU cores and also for future works to be carried out on the LEU core model.

The authors acknowledge the financial support given by the International Atomic Energy Agency (IAEA) and the Zambian Government through the National Institute for Scientific and Industrial Research (NISIR) to carry out this work. The Ghana Atomic Energy Commission (GHAEC) is acknowledged for authorizing access to the Ghana Research Reactor-1 and provision of information that was important for executing this work. The authors also recognise and appreciate the contribution made by members of staff under the Nuclear Engineering department of the Graduate School of Nuclear and Allied Sciences (SNAS) who contributed to the success of carrying out this work.

Parameter | Temperature Range (˚C) | Irradiated Core (mk/˚C) | Clean Core (mk/˚C) |
---|---|---|---|

Fuel Temperature Reactivity Coefficient | 20 - 40 | −0.00347 | |

50 - 100 | −0.00294 | ||

200 - 600 | −0.00266 | - | |

Average (mk/˚C) | −0.00303 | - | |

Moderator Void | 20 - 40 | −0.08644 | |

Reactivity Coefficient | 50 - 100 | −0.19194 | |

Average (mk/˚C) | −0.13871 | - | |

Moderator Temperature (only) Reactivity Coefficient | 20 - 40 | 0.007848 | |

50 - 100 | 0.003924 | ||

Average (mk/˚C) | 0.005886 | ||

Combined Moderator | 20 - 40 | −0.08921 | |

Reactivity Coefficient | 50 - 100 | −0.15456 | |

Average (mk/˚C) | −0.12188 | −0.13 (SAR) |

The authors declare no conflicts of interest regarding the publication of this paper.

Mweetwa, B.M., Ampomah-Amoako, E., Akaho, E.H.K. and Odoi, C. (2018) Prediction of Neutronic and Kinetic Parameters of Ghana Research Reactor 1 (GHARR-1) after 19 Years of Operation Using Monte Carlo-N Particle (MCNP) Code. World Journal of Nuclear Science and Technology, 8, 160-175. https://doi.org/10.4236/wjnst.2018.84014