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# Quantifying gadolinia burnable absorber rods for APR1400 nuclear reactor core

image credit: reactor core and fuel assembly (image from google)

*by Mark Gino Aliperio*

### Introduction

Burnable absorbers (BA) are materials that have a high neutron absorption cross-section that are converted into materials of relatively low absorption cross section as the result of radiative capture. Due to the burnup of the absorption material, the negative reactivity of the burnable absorber decreases over core life. Ideally, these absorbers should decrease their negative reactivity at the same rate the fuel’s excess positive reactivity is depleted. In PWRs burnable absorbers are used to decrease initial concentration of boric acid (also to decrease BOC MTC) and to decrease relative power of fresh fuel assemblies. Fixed burnable absorbers are generally used in the form of compounds of boron or gadolinium, shaped into separate lattice pins or plates, or introduced as additives to the fuel. Since they can usually be distributed more uniformly than control rods, they are less disruptive to the core power distribution.

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In nuclear industry gadolinia (Gd2O3) is commonly used as a neutron absorber due to very high neutron absorption cross-section of two isotopes Gd-155 and Gd-157. In fact, their absorption cross-sections are the highest among all stable isotopes. For this reason, gadolinium is widely used as a burnable absorber, which is commonly used in fresh fuel to compensate an excess of reactivity of reactor core. In comparison with another burnable absorbers, gadolinium behaves like a completely black material. Therefore, gadolinium is very effective in compensation of the excess of reactivity, but on the other hand an improper distribution of Gd-burnable absorbers may lead to unevenness of neutron-flux density in the reactor core.

In this report, the correlation between k-inf at the beginning of cycle, wt% and number of gadolinia BA rods was determined for both 16×16 and 17×17 fuel assemblies. The coefficients of the correlation was obtained by using the Least Square Method. In addition, the change in the k-inf values with respect to varying wt% (4 wt% and 8 wt%) and number (4, 8, 12) of BA rods is discussed. On the other hand, the average value of wt% for a given number of BA rods was determined for APR1400 with cycle length of 17.5 GWD/MTU and average core enrichment of 2.7%, by using the obtained correlation.

### Experimental Design and Methodology

The simulations in this report were done using CASMO3. The source codes for both assemblies (16×16 and 17×17) are provided in the Appendix. The configuration was set for no fuel zoning. For both assemblies, gadolinia rods were used as burnable absorbers (4 wt% and 8 wt%), with boron concentration set at 10 ppm. The average enrichment of the assembly, regardless of the number and wt% of gadolina, was set to be at 2.7%. To ensure this condition, different values of fuel enrichment was used with varying conditions, which can be calculated using the equation (derived using simple average),

where Ntot is the total number of fuel rods (Ntot = 236 for 16×16 assembly; Ntot = 264 for 17×17 assembly); εavg is the average enrichment of the assembly which is 2.7% for all cases; wt%BA is the weight percent of gadolinia in the BA rod; εBA is the fuel enrichment for BA rod which is 0.711%; and NBA is the number of BA rods in the assembly (must be a factor of 4) where in this report, NBA was varied to be 4, 8, and 12 per fuel assembly. The values of fuel enrichment for every condition is presented in Table I, by using the above equation.

The reference state condition was also set to constant where Tf (fuel temperature) = 960.95 K, Tm (moderator temperature) =585.5 K. The location of the gadolinia BA rods for both assemblies are shown in the figure below.

As presented in the figure above, the locations of the BA rods are represented by the number ‘3’ entries in the octant assembly (enclosed in red circle). Moreover, entry ‘1’ represents the fuel rods with enrichment value presented in Table 1. Take note that if the BA rod is located on the diagonal side of the triangle (or even at the vertical), it is multiplied by 4 since it is shared between two octants of the assembly. Otherwise, it is multiplied by 8 (octant). Thus, the total number of rods is determined. In addition, the location of the BA on the assembly is reasonably chosen.

To find the correlation between k-inf at the beginning of cycle (BOC) and wt% and number of BA rods for two assemblies, the proposed correlation was used:

where, w is the wt% of gadolinia BA; r is the number of BA rods; and a0, a1, a2, a3, a4, a5 are unknown coefficients. Now as the initial conditions are set, the simulations are then conducted and the value of k-inf are recorded when burnup is at 0. Since there are six (6) unknown coefficients, a minimum of six equations are also required. The coefficients are calculated using the Least Square Method with a 6×6 matrix where entries are placed as follows:

where, x represents the wt% of BA rods; y is the number of BA rods; and z is the k-inf value for corresponding x and y values. Moreover, a, b, c, d, e, f are the six unknown coefficients; and N = 6 for this report. The validity of the obtained coefficients is then tested if it satisfies the correlation equation with an acceptable error value. Lastly, to determine the total number of BA rods and its corresponding average wt% for APR1400 with cycle length of 17.5 GWD/MTU, the correlation is utilized. This is when k-inf is 1.05 (mentioned in class). It is also assumed that the core is loaded with an average enrichment of 2.7%

### Results and Discussion

Table II presents the different k-inf values for all cases with varying number and wt% of BA rods for both assemblies. It can be noticed that as the number of BA rods in the assembly are increased from 4 to 8 to 12, the k-inf, which is an indicator of the criticality of the reactor, decreases; thus, the inverse proportional relationship. Similarly, when there is more composition of gadolinia in the BA rods (from 4wt% to 8wt%), the k-inf decreases as well. This is obvious as more number and higher wt% of BA, there is more absorption of neutrons, which is attributed to the very high neutron absorption cross-section, σa, of gadolinia for both of its isotopes (Gd-155 and Gd-157). Moreover, similar trend of results is observed in the 17×17 fuel assembly, although, the k-inf values are a bit higher with that of the 16×16 fuel assembly.

Using the values of the k-inf presented in Table II, the coefficients of the correlation from the above equation are calculated using the Least Square Method. The results are presented in Table III for both assemblies.

As presented in Table 3, the validity of the obtained coefficients is tested, and the values are plugged into the correlation equation. The comparison of the values from CASMO simulation results and Least Square Method (LSM) calculation, and its corresponding error is presented in Table 4. Consider for example the case of 4 BA rods with 4 wt%, that is:

As shown in the Table IV, the results from CASMO simulation compared to the correlation equation using LSM has error values of 0.01295% and 0.00482% for 16×16 and 17×17 fuel assemblies, respectively. Thus, with this negligible amount of error, it is safe to say that the obtained coefficients of the correlation equation using LSM are valid and accurate.

Using the correlation equation and the values of its corresponding obtained coefficients, the total number and average wt% of BA is calculated for APR1400 cycle length of 17.5 GWD/MTU with core average enrichment of 2.7%. Take note that for such condition, the k-inf value is at 1.05 as determined by the results of the first homework. Since the correlation is a second-order equation and has only two variables, setting the value for r for example, yields to a quadratic equation. This is easily calculated using the quadratic formula. The results of the computations are shown in the table below:

The table above shows the determined approximate wt% of BA rods for a given number of BA rods for APR1400. When there are four (4) gadolinia BA rods in the assembly, wt% is found to be 3.33809. When the number of BA rods in the assembly is increased to eight (8), its corresponding wt% is found to be 7.23512. Take note that this wt% for number of BA rods satisfies the condition of 17.5 GWD/MTU cycle length, core average enrichment of 2.7%, and 10 ppm Boron concentration.

### Conclusion and Summary

In this report, the correlation between k-inf at the beginning of cycle, wt% and number of gadolinia BA rods was determined for both 16×16 and 17×17 fuel assemblies. The effect on the k-inf when wt% and number of BA rods were varied was presented. It is found that increasing number or increasing wt% of gadolinia BA rods decreases its corresponding k-inf. Moreover, the k-inf values for 17×17 assembly is higher compared to 16×16 fuel assembly.

On the other hand, from the correlation, the average wt% was approximated for a given number of BA rods for the case of APR1400 with 17.5GWD/MTU cycle length with core average enrichment of 2.7%

*This report on quantifying gadolinia burnable absorbers, has been conducted in partial fulfilment of the requirements of the course EN304 Nuclear Design Codes and Methodologies.*

*References:*

[1] http://www.nuclear-power.net/nuclear-power-plant/nuclear-fuel/burnable-a... [2] J. Lamarsh & A. Baratta, “Introduction to Nuclear Engineering 3rd ed”, Prentice-Hall Inc (2001) [3] http://calculator.vhex.net/calculator/linear-algebra/least-squares-solution

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