Calculating Weight-Loss in Nuclear Graphite
Weight-loss by radiolytic oxidation is a life-limiting factor in the graphite cores of Advanced GasCooled Reactors (AGRs), as too much weight-loss can compromise the moderating abilities and
structural integrity of the nuclear reactor. As the overall weight-loss cannot be physically measured,
models have to be made to calculate and predict for how long the reactors can continue operation.
A physically based set of baseline predictions, forecasted by a program called FEAT, form the basic
model and these are scaled using a number of calibration factors to create more specific models. These
are formed by multiplying FEAT predictions by attack ratio, which is the experimentally determined
weight-loss found on samples cut or ‘trepanned’ from the reactor core, divided by the scaled predicted
values.
The weight-loss predictions have recently been re-calculated for Hinkley Point B and Hunterston B
power stations and it was my job to independently assess and re-forecast them.
Figure 1: a trepanned sample from the graphite core
Attack Ratio
[Note: values have not been shown in calculations for confidentiality reasons]
First I had to find an appropriate model for attack ratio.
I had to average the observed and predicted weight-losses for 295 trepanned samples which had each
been cut into six slices and then divide them by each other to find attack ratio, using the equation:
n
x
Attack Ratio =
i 1
n
y
i 1
Where: x = observed weight-loss for each slice in a sample
y = predicted weight-loss for each slice in a sample
I then had to plot these attack ratio values against core burnup, which is a measure of channel
irradiation, to find an appropriate model.
I found several fits calculated by choosing variable parameters and minimising the residuals, as follows.
HPB All, Core Burn-Up v Average Weight-Loss
For example, to find an exponential fit, I
chose the form:
35
30
Predicted attack ratio = a +becx
Weight-Loss (%)
25
20
15
10
5
0
0
2000
4000
6000
8000
10000
12000
14000
16000
Core Burn-Up (GWd)
Figure 2: core burnup v average weight-loss from the old models (weight-loss values
not shown for commercial reasons)
Where a, b and c are variable
parameters. To find the fit, I had to
minimise the sum of the square of the
residuals, which are calculated by
(predicted attack ratio – actual attack
ratio)2. Due to the large dataset and
number of parameters, this is done using
the ‘solver’ function on Microsoft Excel,
which uses an iterative formula to
minimise the residuals by changing
the variables.
The goodness of the various fits relative to each other must then be determined. I did this by using the
Akaike Information Criterion (AIC), calculated as follows.
First find the standard deviation:

1 n
xi   2

n i 1
Where: σ = standard deviation
µ = model prediction
n = number of values
Then find the log-likelihood:
n
n
1
ln( p)   ln( 2 )  ln( 2 ) 
2
2
2 2
n
 (x
i 1
i
 )2
These are derived from the normal distribution.
AIC can be calculated:
AIC  2K  ln( p)
Where: K = number of parameters
The AIC penalises for poor fitting of the data and for over-parameterisation of a model, so the lower the
AIC value, the better the model.
Using the attack ratio prediction model with the lowest AIC value, new weight-loss predictions can be
made by multiplying the FEAT baseline values by the predicted attack ratios. Doing this allows us to
make safety cases and continue the safe operation of our nuclear reactors.