Doppler coefficient

Fourth, for comparable reactor systems, the one using a thorium-base fuel will have a larger negative feedback on neutron multiplication with increased fuel temperature (Doppler coefficient) than will a U-fueled reactor. 

Spectral effects are probably also important for integral parameters other than material reactivity worth. An example is the Doppler coefficient, which is caused by resonance perturbations. No satisfactory agreement has yet been reached between the calculated and measured Doppler effect of small fissile samples in fast assemblies 23). 

Figure 4.2-6 shows the calculated temperature coefficient of reactivity for the BOC-IC condition. Curve A is the fuel prompt doppler coefficient due to heatup of the fuel compact matrix as a function of the assumed fuel temperature. Curve B is the active core isothermal temperature coefficient and is the Siam of the doppler coefficient and the moderator temperature coefficient of reactivity which is also strongly negative, due in large measure to the presence of LBP in the BOC condition. The moderator coefficient, not shown in Figure 4.2-6, would be the difference between Curve B and Curve A and would be -4.0 x 10" / C at 800 C (1472 F), for example. Curve C is the total reactor isothermal coefficient and includes the positive contribution of the reflector heatup to the estimated inner and outer reflector temperatures that would result when the fuel reaches the indicated temperature. 

The Doppler effect in the fuel would be expected to stabilise the void coefficient with a negative effect. But the Doppler coefficient only operates if the fuel temperature is high. Operated below 30% power, the normally high temperature rise in the ceramic fuel is much decreased and the Doppler effect inoperative until, during the course of the accident, the fuel temperature rose markedly. Indeed it is possible that this was the mechanism that terminated the nuclear excursion before break-up. 

It is well recognized that the Doppler coefficient is one of the most important counter-reactions during reactivity excursions. The increase of the fuel temperature causes an increase in the amplitude of the uranium-238 neutron capture resonances and, therefore, a decrease in the core reactivity. In some transients, it is conservative to assume a most negative Doppler coefficient (when a higher power... 

The Doppler coefficient varies with the fuel burn-up, that is with the operation time, becoming less negative (i.e. less effective as a safety counterreaction) when the bum-up increases. In fact, with time, four phenomena cause a variation of the coefficient ... 

The last factor predominates over the others and, at the end of the core life, the Doppler coefficient is less effective. The two curves in Figure 4-2 to be used for transient analysis, are the result of the fuel burn-up and the uncertainties of evaluation. 

In general, it is possible to ensure that the additional reactivity due to a control rod expulsion is of the order of 0.15 per cent (but, in any case, well below 0.6 per cent, which would originate a prompt criticality ). The accident reactivity excursion is mitigated by the Doppler coefficient and is terminated by the reactor scram. Roughly 10 per cent of the fuel can be damaged (DNBR < 1) and the effective whole-body doses outside the plant may reach 10-20 mSv in two hours at the edge of the exclusion area. 

Highly enriched mixed oxide (MOX) fuels and Pu fuels without uranium were considered for Pu burning enhancement. It was found that Pu consumption rates essentially depend on Pu enrichment. Both bumup reactivity loss and Doppler coefficient are important criteria for highly enriched MOX fuel cores. Cores without uranium were found to consume the Pu at a very large bumup rate close to the theoretically maximum value of 110-120 kg/TWhe. The introduction of UO2 in an internal blanket is effective in enhancing the Doppler coefficient, it causes a minor increase in the sodium void reactivity in non-uranium cores. 

Feasibility studies have been performed to investigate the basic characteristics (transmutation rate, bumup reactivity, Doppler coefficient, sodium void reactivity, maximum linear heat rate, etc.) of a fast reactor core with MA transmutation, the following items were considered ... 

The significance of these predictions is shown in Table XV. If the nuclear data are adjusted in line with this experimental information, the design calculations show that at least 50% of the BeO, which was included in the core to soften the neutron spectrum and to increase the Doppler coefficient, can be removed and still meet the Doppler and sodium loss criteria of the reference core. The fissile inventory required decreases by about 3%, and the breeding ratio increases from 1.18 to about 1.25. This results in a decrease in the fuel cycle cost of about 0.1 mill/kW-hr. If one assumes a favorable combination of nuclear data within the limits of uncertainties reported by Greebler (5) and, furthermore, if the safety criteria are relaxed to allow a calculated Doppler effect (T dkjdT) of — 0.004 (with sodium out) and a positive total sodium loss reactivity effect between 1 and 2, all of the BeO can be removed... 

In the event the isotopes are at different temperatures, a definition of this form is a calculational necessity. But we show that it is also the most convenient definition for calculational purposes even when the temperatures are all equal, because it fortunately results in individual isotopic contributions for the total Doppler effect which are to a high degree of accuracy functions only of the individual isotope s resonances. Thus, in the statistical approximation the Pu- Doppler coefficient is very nearly independent of the resonances in etc.l This is not true of some other possible definitions of the individual isotopic contributions to the total Doppler effect for a mixture. 

Some representative numerical results on Doppler coefficients, effective cross sections, and auxiliary functions used in the calculation of Doppler coefficients are quoted and discussed in this section. Furthermore, a brief survey of the more unusual methods, such as the Monte Carlo technique, is given. We do not attempt to review all the work. For a description of additional calculations, the reader is referred to the excellent review paper by Nordheim (75). 

Almost all of the practical calculations of the Doppler effect in fast reactors are carried out by the conventional method, which consists of tracing through the expressions for the effective cross sections, as developed in this paper for the different fuel isotopes and energy ranges. Then, one can either run two multigroup diffusion calculations with the effective cross sections at different temperatures and obtain the reactivity changes associated with the Doppler effect as the difference of the two criticality factors, or else one can do only one multigroup calculation and find the Doppler coefficient by the use of perturbation theory. Both ways are, of course, equivalent from a mathematical point of view. 

The results to be quoted were all obtained by one of these methods. The calculated Doppler coefficients, which have been published for many fast breeder designs, are for systems that differ widely in size and shape of core and blanket, in fuel composition, core composition, etc. A comparison of many of them would not be meaningful, and so a few values will be selected which are suitable to demonstrate the major trends. 

The Doppler coefficients in Fig. 2 refer to Pu02-U(238)02-fueled, sodium-cooled fast breeder reactor systems. They all have negative Doppler coefficients, because the cores contain substantially more fertile than fissile material. The abscissa in the figure shows the ratio of fertile to fissile nuclei, or + Pu ) over (Pu Pu " ), which is... 

Fig. 2. Calculated Doppler coefficients. See Table IX for the significance of the data points.

Besides the standard method of calculating Doppler coefficients, there are two somewhat unusual approaches described in the literature. Both of them are set up such as to be independent of all the approximations made in the standard method, and they can treat effects like heterogeneity, resonance overlap, multilevel fits, etc., at the expense of long computer times, and both of them require high-speed digital computers. One of them is a purely numerical approach, which is being developed in the... 

P. Greebler, B. A. Hutchins, and J. R. Sueoka, Calculation of Doppler coefficient and other safety parameters for a large fast oxide reactor. GEAP-3646 (1961). 

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