How to explain the effect of temperature profile on feedback coefficient?

[mark_bose]

Hi all,

I'm trying to derive fuel temperature coefficient in a TRIGA reactor using a monte carlo code. When i do that, if i assume a radial temperature profile along the core, i obtain smaller value (-7pcm/K) than the one achieved with uniform temperature (-9pcm/K).

More in detail: in my case α_f is negative and decrease with temperature, since in case of a radial temperature profile the absolute value of α_f is smaller, i thought that the effective temperature of the core was smaller then the average one, this would mean that external fuel rods have an higher importance then inner rods. But if it is the case, how could i explain this from a phenomenological or mathematical point of view? what should i look for?

Did you know any paper or book that cover this topic?

I appreciate every suggestion

Thanks.

 

[rpp]

Can you provide a few more details on how you are calculating your Doppler coefficient?
I assume you are starting with a full-core MC calculation at one temperature, running a full-core MC calculation at another temperature, and calculating the Doppler coefficient.
1. What is the temperature and profile you are using at your initial state?
2. What is the temperature and profile you are using at your final state?
3. What MC code are you using that gives cross sections at arbitrary temperatures?
4. What is the uncertainty in your MC k-effective calculation?

The answer is going to depend on how large a temperature change you are making, whether the change is uniform throughout the core or proportional to power, and what the distribution is before and after.

There is also a question of if your MC can calculate "on-the-fly" temperatures, and whether you are converging your MC answers tight enough. Using rough numbers, if your temperature change is 10 degrees and your Doppler is -5 +/- 0.5 pcm/K, then the keffective delta is 50 +/- 5 pcm. However, you are taking the difference between two codes, so each code needs an uncertainty of 2.5 pcm. If you want 2 sigma significance, it is even harder and your MC codes need to be converged to about 1.25 pcm. Calculating reactivity coefficients with MC codes can become difficult. (These are very rough numbers).

 

[Alex A]

Really solid points by @rpp . In addition to those it turns out if you scratch this problem on the surface, underneath is a screaming nightmare quite a worthwhile challenge.

https://ansn.iaea.org/Common/docume...tors (Safety and Technology)/pdf/chapter1.pdf

That document has a number of very non classical things to say about zirconium hydride based moderator/fuel, and credits them with providing the reactivity curve.

If you need a mathematical insight, I would suggest calculating the Boltzmann population at thermal, 0.14ev and maybe even 0.28ev. I would try calculating how this changes the fate of a neutron, either with fission and capture tables or by MCing a reactor with a thermal and then with a 0.14ev source (and no moderator).

 

[mark_bose]

I'm using Serpent and the assumption of @rpp about the procedure is correct. For what concern temperature profiles I'm using an exact proportionality with power in each ring of the core (labeled as "B" "C" "D" ...).

Initial and final T. profiles are taken starting from power distribution and then fixing the desired average temperature. The minimum temperature variation is in the outer ring and it's about 20 degrees, the average temperature variation is around 50 degrees at each step.

The uncertainty on k-eff is around +-0.00015. Below you can see the effect of temperature on k.

The effect is quite big compared to the uncertanty, so i would say that is not a matter of uncertainty or convergence, what's your opinion?Thanks @Alex A for the document. I tried to do something similar to your suggestion, i evaluated the total amount of fission and absorpion in each region of the core. It seems that in the outer region the difference (#Fission-#Capture) is higher than in the inner region, i tried also to compute an effective temperature using as weight this difference. The result is an effective temperature which is smaller than the average one (580K against 610K) but in my opinion it is not sufficient (Feedback coef. at an average T of 580 K is again much higher).

 

[rpp]

Sorry for the delay in answering. The short answer is that differences that you are seeing look reasonable. The temperature distribution is going to make a difference on the Doppler coefficient.

This effect is often observed when looking at transient analysis. During a transient, the temperature distribution inside the fuel rod is going to be different than steady-state operations and can affect the temperature coefficient. In transient codes, one approximation is to model an "effective" temperature which is the weighted average of the centerline (or average) and surface temperatures like
$$ T_{eff} = \omega T_{avg} + (1-\omega) T_S $$
where a value of $$\omega=0.92$$ has been determined from Monte Carlo calculations [1].

Another value used in NEA benchmark problems (originally from Westinghouse?) is [1]
$$ T_{NEA} = 0.7 T_S + 0.3 T_C $$
Note that this uses the centerline temperature, not the average.

It might be interesting to see if the value of $$\omega$$ is close to what you have observed.[1] G. Grandi, K. Smith, Z. Xu and J. Rhodes, "Effect of CASMO-5 Cross-Section Data and Doppler Temperature Definitions on LWR Reactivity Initiated Accidents". PHYSOR 2010, Pittsburgh, PA, (2010)