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The problem is concerning the use of a thermal fluxed squared weighting factor in a thermal reactor.

I have seen in sources the thermal flux in a reactor is squared as a statistical weighting factor, for example looking at temperature feedback. Say a number of thermocouples are measured in a reactor, the weighting of the importance of the thermocouples are to flux squared. Why is this when the reaction rate is equal to the flux? Is it simple statistics I'm not understanding.

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$$ \Delta \rho \cong -\frac{\int_V \! \mathrm{d}^3r \phi(\mathbf{r}) \Sigma_a(\mathbf{r}) \phi(\mathbf{r})} {\int_V \! \mathrm{d}^3r \phi(\mathbf{r}) \Sigma_f(\mathbf{r}) \phi(\mathbf{r})} $$

It seems like because of a result of one group perturbation theory. Page 223 of J. Duderstadt Nuclear Reactor Analysis. Since the temperature feedback is changing the thermal utilisation, which is the ratio between the absorbed and utilised thermal neutrons.

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Well lets examine the situation that your are describing. When talking about anything 'thermal', one is referring to 'heat'. Heat can come in at least two forms that I am aware of. Radiative and Kinetic. Photons, of a thermal nature, come from the infrared spectrum of light. While any photon can impart energy into a material, it has to be provided that the material absorbs the photon. Temperature is a measure of a rate of Kinetic Energy transferred from a system high in KE to a system of lower KE. Matter vibrates, or rather the molecules in matter vibrate. The level of this vibration can be measured as Black-body Radiation. Objects in the same system, at equilibrium, if they are not active sources, will emit the same level of Black-body radiation. If there is an active source of light, that is illuminating the non-active objects, they will emit differing levels of Black-body Radiation, due to any differentiating material qualities.

All objects, that have form, fit and function, have a surface. The 'Flux' of anything is related as a function of surface, a 2D concept. As an example of this is the Magnetic Flux of the surface of a magnet. The more lines of force per unit of surface area, the stronger the magnet. The number of lines of force coming in and out of the surface of the magnet is the Magnetic Flux. In the case of thermal photons, it would be the number of photons hitting the surface of a sensor, such as a thermocouple, per some unit of time, usually a second. In the case of a thermometer, such as a Mercury thermometer, the Kinetic energy 'flux' of the material being measured, such as boiling water, in contact with the surface of glass that makes up the body of the thermometer, transmits that KE through the glass and into the mercury, which then expands, dependent upon the rate of transfer. The transfer rate, equal to the flux, is dependent upon the rate of the generation of what makes-up the 'flux', be it boiling water, photon density, or thermal neutrons.

Often one has a volume of something to content with, and measurements are often taken on a cross-section of that volume, the cross-section is a 2D concept. A single thermocouple, as a source of measurement, is considered a point concept. If you have at least three of them in a volume, then you can define a triangle of surface within the volume, a cross-section if you will, thus a 2D concept. Any distance squared will give you a function of area, a 2D concept, so the flux is squared as well.

Δρ≅−(∫ V d 3 rϕ(r)Σ a (r)ϕ(r)) / (∫ V d 3 rϕ(r)Σ f (r)ϕ(r)) is given as a function of volume, and is given as a unit-less ratio, but could be considered as a difference in 'degrees' just like the lines on the thermometer, indicate a change in volume of the mercury, thermocouples give a difference in electrical output in millivolts, which get translated into a temperature. Typically here you have 2 cross-sections, each having a given flux, and there is a difference of flux between each cross-section, but still is referenced as an 2D concept, even though one is taking the temperature, say in degrees, a degree is not easily translated into actual units of distance, time, or mass.

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