On a number of occasions in this weblog I have talked about the central importance of the temperature coefficient of reactivity in the safety of nuclear reactors, because the temperature coefficient governs how a reactor responds to changes, transients, accidents, etc. It must ALWAYS be negative, and strongly negative temperature coefficients are even better.

While reviewing some of the old ORNL documents I was able to compare and contrast the difference in temperature coefficients between the one-fluid design and the two-fluid design. The information and relevant sections are reproduced here, and the parent documents are also available on this site.

From pages 68-70 of ORNL-4528: Two-Fluid Molten-Salt Breeder Reactor Design Study (PDF, 6.9M)

6.5 Temperature Coefficients of Reactivity

In analyzing power transients in the two-fluid MSBR, one must be able to determine the reactivity effects of temperature changes in the fuel salt, the fertile salt, and the graphite moderator. Since the fuel is also the coolant and since only small fractions of the total heat are generated in the fertile salt and in the moderator, one expects very much smaller temperature changes in the latter components than in the fuel during a power transient. Expansion of the fuel salt, which removes fuel from the active core, is thus the principal inherent mechanism for compensating any reactivity additions.

We accordingly calculated the magnitudes of the temperature coefficients of reactivity separately for the fuel salt, the fertile salt, and the graphite over the range of temperatures from 800 to 1000K. The results of these calculations, as shown in Fig. 6.10a, illustrate the change in multiplication factor vs moderator temperature (with ?k arbitrarily set equal to zero at 900K). Similar curves of ?k vs temperature for fuel and fertile salts are shown in Figs. 6.10b and 6.10c, and the combined effects are shown in Fig. 6.10d. All these curves are nearly linear, the slopes being the temperature coefficients of reactivity. The magnitudes of the coefficients at 900K are shown in Table 6.8.

The moderator coefficient comes almost entirely from changes in the spectrum-averaged cross sections. It is particularly worthy of note that the moderator coefficient appears to be quite insensitive to uncertainties in the energy dependence of the 233U cross-sections in the energy range below 1 ev. This is to say that reasonable choices of cross-sections based on available experimental data yield essentially the same coefficient.

The fertile salt reactivity coefficient comprises a strong positive component due to salt expansion (and hence reduction in the number of fertile atoms per unit core volume) and an appreciable negative component due to temperature dependence of the effective resonance absorption cross sections, so that the overall coefficient, though positive, is less than half as large as that due to salt expansion alone.

The fuel salt coefficient is due mainly to expansion of the salt, which of course reduces the average density of fuel in the core. Even if all core components were to undergo equal temperature changes, the fuel salt coefficient would dominate. In transients in which the fuel temperature change is far larger than that of the other components, the fuel coefficient is even more controlling.

Now from pages 63-64 of ORNL-4548: Molten-Salt Reactor Program: Semiannual Progress Report for Period Ending February 28, 1970 (PDF, 57.0M)

6.13 Reactivity Coefficients

A number of isothermal temperature coefficients of reactivity were calculated for the single-fluid MSBR, using the reference reactor geometry shown in a previous progress report. These calculations were performed with a detailed two-dimensional representation of the reactor in R-Z geometry, using the diffusion code CITATIONs with nine neutron energy groups. Both forward and adjoint fluxes were calculated, and the effects of various changes in microscopic cross sections or in material densities were calculated by first-order perturbation theory. The cross sections themselves were obtained from a series of calculations, using the code XSDRN in which group-average cross sections were calculated for each major region of the reactor for each of three different temperatures (800, 900, and 1000K) and for various combinations of material densities. In this way the effects of temperature-dependent changes in microscopic cross sections can be calculated separately from those of temperature-dependent changes in density.

The calculated reactivity coefficients are summarized in Table 6.3. The Doppler coefficient is primarily that of thorium. The graphite thermal base coefficient and the salt thermal base coefficient, that is, the effects of microscopic cross-section changes caused by changing the temperatures of the graphite and the salt, respectively, are positive because of the competition between thermal captures in fuel, which decrease less rapidly than those of a l/v absorber, and thermal captures in thorium, which decrease nearly as l/v, with increasing temperature. The salt density component represents all effects of salt expansion including the decreasing salt density.

The graphite density component includes both changing graphite density and displacement of graphite surfaces. In calculating the displacements it was assumed that the graphite-vessel interface did not move, that is, that the vessel temperature did not change. For short-term reactivity effects, this is the most reasonable assumption, since inlet salt bathes the vessel’s inner face. These dimensional changes in the graphite without a concomitant expansion of the vessel produce a significant change in the thickness of the salt annulus between the core and the reflector. The reactivity effect of this change is not readily calculated by perturbation theory and was therefore obtained by comparison of two conventional criticality calculations with different thicknesses of the salt annulus and with appropriately differing core density. In any case, it should be noted that the graphite density coefficient is a small and essentially negligible component.

From Table 6.3 it is seen that the total core coefficient is negative. But more important, the total salt coefficient, which is prompt and largely controls the fast transient response of the system, is a relatively large negative coefficient and affords adequate reactor stability and controllability.

The salt density coefficient is particularly important with regard to bubbles in the core salt. It is expected that the salt will contain about 1% helium bubbles. Under certain circumstances the bubbles might expand or collapse without change in core temperature and hence without invoking the total sa
lt temperature coeficient. Since the salt density component is positive, bubble expansion would produce a positive reactivity effect. Using a salt expansion coefficient ?V/V = 2.1 x 10^-4/°C, an increase in core bubble fraction from, say, 0.01 to 0.02 would yield a reactivity change of ?k/k = +0.00039. This is approximately one-fourth the worth of the delayed neutrons in the core. Analogously, complete collapse of a 0.01 bubble fraction would yield a reactivity change of ?k/k = -0.00039.

Finally, the fuel concentration coefficient, (?k/k)/(?n/n), where n is atomic density, was calculated to be 0.42 for 233U and 0.027 for 235U. The large difference between these two numbers is primarily a result of the substantial difference in concentrations (i.e., n23 ? 11 x n25), so that a given fractional increase in 235U concentration produces a far smaller reactivity effect than does the same fractional increase in 233U concentration.

As can be seen from the data, the isothermal temperature coefficient for the two-fluid reactor is roughly five times as strong as the isothermal temperature coefficient for the one-fluid reactor (-4.34 vs -0.87). The isothermal temperature coefficient refers to the situation where a temperature change has made its way evenly throughout the reactor. Since most temperature disturbances originate in the fuel, the fuel temperature coefficient is even more important, and in this respect the 2-fluid is over twice as strong as the 1-fluid (-8.05 vs -3.22).

There is even recent work that casts doubt on whether a one-fluid LFR has a negative temperature coefficient at all!

These strong differences between the magnitudes of the temperature coefficients show how the two-fluid LFR design has the potential for greater safety and margin than the one-fluid design later favored by ORNL.