Does the following formulation have merit?

[BSFusion]

I recently came across an interesting article, titled “Analysis of an Improved Fusion Reaction Rate Model for Use in Fusion Plasma Simulations” (fdm1268.pdf), published in 2005 by the Fusion Technology Institute. The article compares a handful of mathematical formulations of , in terms of computational speed and accuracy. The authors determined that the overall best approach was a very complicated model, by H. S. Bosch and G. M. Hale, using R-matrix theory.

After I finished reading that article I was left wondering, why are all of these mathematical models so messy? Is it possible to have a simpler formulation, one that would permit integration using elementary Calculus techniques? That way, some useful information could be calculated directly, avoiding the hassle of having to program and run time-consuming simulations.

The reason this interested me was because I have an invention, Bubble-confined Sonoluminescent-laser Fusion (BSF), which needed to have its method of ignition validated. Specifically, I required a model of BSF using equimolar DT between 0.2 – 10 keV, and, as far as I could tell, there were no mathematical models designed to fit this low-temperature self-heating regime.

My preliminary investigation, toward creating a simplified fusion reaction rate formulation for use in fusion plasma calculations, was successful, and quickly resulted in the following polynomial expression, with T in keV and fitting constants C0-C8, in units of 10^-23 cm^3 n^-2 s^-1:

POLYFIT = C0 + C1*T + C2*T^2 + C3*T^3 + C4*T^4 + C5*T^5 + C6*T^6+ C7*T^7 + C8*T^8.

C0 = -1.3836835 C3 = 1427.2571 C6 = -931.66276
C1 = 32.377050 C4 = -3490.1717 C7 = 92.414366
C2 = -302.43936 C5 = 3849.6795 C8 = -3.3327230

A comparison can be made, using the chart below, based on an equimolar mixture of D & T. Notice, at low-temperatures POLYFIT has, by far, the best fit of the group. In addition, by adding two more fitting constants, the relative error of POLYFIT can be reduced even further, making it fit either BUCKY or Bosch-Hale graphs, so it varies from them by less than 1% throughout the entire 0.2-10 keV range.

—————————————————————-
Various formulations of compared to Bosch-Hale
—————————————————————-
Temperture NRL DRACO BUCKY POLYFIT
(keV) (Rel%Err) (Rel%Err) (Rel%Err) (Rel%Err)
—————————————————————-
1 19.79 20.08 20.03 1.89
2 12.68 12.00 12.16 2.72
5 4.82 4.82 4.97 1.09
10 3.18 4.94 5.10 6.06

The fitting constants for POLYFIT were calculated using a standard least squares polynomial fitting technique, in such a way that when the individual relative-errors (difference between predicted and observed, relative to observed) were squared, and those values accumulated, the total was a minimum. As good as this method is, it can still be improved. For example, instead of limiting the expressions to natural number exponents, negative and/or fractional exponents could be tried. In addition, why should the POLYFIT polynomial be based upon mimicking another, possibly inferior, formulation, like Bosch-Hale? It makes more sense to start from scratch with all the raw experimental data, which can then be weighted in accordance with the certainty of its experimental error-bounds.

Does this make sense, or am I confused?

[vansig]

trouble is, while it’s a fairly straightforward matter to write differential equations, the kinks in plasmas are regions where the circumstance is not exactly laminar, and chaotic fluids do not solve easily with normal methods.

unless of course, you can make quantifiable predictions that are supported by experiment?

[asymmetric_implosion]

I view the problem as one of v, velocity, in the . If the ion distribution is Maxwellian, the problem is pretty straightforward as you described. The real problem is when the ion distribution must be calculated. If you know you have a Maxwellian distribution, you can go forth with relative ease. The problem in Tokamek, Z-pinches fusion and other plasma based systems is the ions can have bumps on the tail of the distribution. I am not familiar with Sonoluminescence based fusion techniques but my guess is they have a thermal distribution due to a large collision frequency and the absence of a magnetic field. The laser, if powerful (>1 TW), could lead to fast ions but you need to do some calculations on the thermalization time of the ions. If the thermalization time is short compared to other time scales like the applied power pulse or some characteristic lifetime, you can assume a Maxwellian distribution with reasonable accuracy. Other sources of fusion are not so lucky in a computational sense.

One could argue that the fusion cross section is not taken in fine enough increments but that is the typical problem of cross section data. More could be done but the gains are viewed as small at this time.

[KeithPickering]

BSFusion wrote:
Does this make sense, or am I confused?

Considering that the coefficient for the highest power of T is negative, this is almost certainly a case of statistical overfitting. (Extrapolate the derived curve out far beyond your data, and you’ll see what I mean.) Do you have a [em]scientific[/em] reason for suspecting that fusion rate is a function of T^8?

[BSFusion]

Thanks for the comments. 🙂

Yes, you are correct, but that curve was neither intended to be extrapolated beyond its 0.2 – 10 keV temperature range, nor was it to be used for scientific insight – It was only designed to be a resonably accurate fit over a limitted range of experimental values. The goal, as I stipulated it in my original post, was to find an expression for that would allow easy mathematical manipulation, focusing primarily on ease of integration. If you want a scientific model, the Bosch-Hale formulation can be used, but B-H cannot easily be integrated, and it is so complex that writing it down on paper requires almost a page, and the calculation needs to be broken into three parts prior to performance.

Unfortunately, as vansig & asymmetric_implosion correctly pointed out, the formulation is based on a stable, thermal (Maxwellian) plasma, so applying it outside of those parameters is not appropriate.

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