Doppler feedbacks

The nuclear reactor kinetics was modelled using simple point kinetics. The point kinetics model utilised in the calculation was developed as an analogue to the point kinetics module of the RELAP5 code. The number of delayed neutron groups considered was six. A Doppler feedback coefficient of -0.0095 was used. Xenon feedback was also modelled, although due to the time scales considered in this document the xenon feedback is not relevant and has almost no impact on the results. 

This paper presents the results obtained with the ERANOS calculation scheme and data for several major parameters of the Super-Phenix start up core (CMP) including the critical mass, the control rod reactivity worth, the power map distribution, the burn up reactivity swing and the Doppler feedback. Agreement of calculation results with experiment are very satisfactory agreement within 100 pcm for the critical mass, less than 5% discrepancy on control rod worth, a residual radial gradient of less than 5% the radial power map, an discrepancy less than 10% on the reactivity swing, and a full agreement on the Doppler constant. 

Between the time at which positive reactivity is inserted and control rods begin to move, the reactor behavior depends primarily on the Doppler effect to limit the severity of the accident. The full-power operating conditions correspond to the lowest Doppler feedback in normal operation because the Doppler effect is approximately inversely proportional to the absolute fuel temperature. Also, energy required for core failure is normally a minimum at full core power. Thus, the core is most vulnerable to reactivity insertions at full power. The ejection of the most reactive control rod was assumed at full power. Worth of the most reactive control rod is 60. An instantaneous ejection of this rod was assumed. 

The Doppler feedback during the excursion is then obtained by substituting Eqs. (A2.10) and (A2.12) into the following expression and integrating with respect to time ... 

A spherical version of the basic equations used in the modified Bethe-Tait calculations is summarized below. The simplicity of this model (compared to the cylindrical model) makes it especially attractive for doing parameter studies. In addition, the results obtained with the spherical model compare favorably (see Section V) with those obtained using the more complicated cylindrical model. The set of equations given below includes a heat of fusion correction for the Doppler feedback calculations and a calculation of the explosive energy release. (The latter calculation is not coupled directly to the total energy calculation.)... 

The Doppler feedback calculation is the same in the spherical model as in the cylindrical model [Eq. (A2.8)], except the expression for the average energy density, (and its derivative dQJdi) is in spherical form and contains a heat of fusion correction ... 

The Doppler feedback, however, is large over this range and causes a sizeable reduction in the energy release from a meltdown accident (see Section IV.A). The temperature dependence of the isothermal Doppler coefficient has been found to be very nearly T (4B, 5B). To calculate the Doppler feedback during an excursion, a relation must be known between fuel temperature and the energy density that is obtained from the solution of the point kinetics equations. The relation between energy density and temperature for these conditions is... 

The moderator expansion characteristic of LWRs is not present in the fast spectrum sodium-cooled reactor yet negative reactivity feedback due to thermal expansion prevails. Doppler feedback is generally the fastest acting feedback mechanism since it is almost instantly affected by core power level. The Doppler feedback inserts negative reactivity into the system as the temperature rises and can thus help limit the extent of power increases. However, as fhe fuel temperature drops with a power reduction, the Doppler feedback adds reactivify which tends to limit the power decrease of the system. 

For the transient overpower event without scram, it is assumed that all nine rods would be withdrawn from fheir normal full power posihon at the maximum withdrawal rate. Initially, there would be a rapid rise in power, which is halted by negative reachvity feedbacks from radial core expansion, fuel assembly bowing, and Doppler feedback associated with rise in sodium and fuel temperatures. The PRISM core and fuel can safely accommodate this transient without melhng of fuel or sodium boiling and void formahon in the cooling channels that is, the consequences are benign. 

There are a number of processes that affect thermal feedback, the most important of which are the Doppler feedback coefficient and the steam voidage coefficient. 

Doppler feedback is a key process in reactor stability that takes place at the level of the atomic nucleus. In nuclear reactors, Doppler feedback is due to thermal vibration of uranium-238 nuclei, which makes the nuclei appear bigger to passing neutrons. Hence neutrons are more readily captured by uranium nuclei when the fuel temperature increases. Nuclear reactor fuel is normally only 2.5-3% fissile (heat-producing) uranium-235. The other 97-97.5% is uranium-238, which is non-fissile -i.e., it does not fission easily to produce heat. Instead, uranium-238 captures neutrons and becomes uranium-239, which with a half-life of just over two days becomes plutonium-239 (which is again fissile). The key point is that the immediate effect of higher fuel temperature is that non-fissile uranium-238 mops up more neutrons within the reactor core without producing more heat. 

Hence, Doppler feedback in nuclear reactors works like this. As the reactor fuel becomes hotter, the uranium-238 absorbs more neutrons. This means that, if the reactor fuel becomes hotter, the reactivity reduces. Also, because uranium-235 (which generates the heat of fission) and uranium-238 are mixed at the atomic level, the Doppler Effect is very fast-acting. In essence, the Doppler Effect is the reason we... 

Coolant bulk boiling led to a rapid rise in power, which Doppler feedback could not counteract. 

To enhance the negative reactivity feedback at elevated temperatures, gas expansion modules (OEMs) were added at the core periphery to compensate for the large positive Doppler feedback associated with the decreasing fuel temperature of oxide cores during fission shutdown. The gas expansion modules have a vapour space which will expand with loss of core inlet pressure to increase the core neutron leakage. 

The Doppler feedback coefficient results from any increase in power that generates an increase in fuel temperature. At higher temperatures, uranium atoms have an increased probabihty of absorbing a neutron due to their higher energy state. As the neutrons are absorbed, fewer are available to cause fission. This increase in fuel neutron absorption is referred to as a Doppler effect and helps stabilize the core if there is an unexpected increase in temperature. 

The fuel rod heat transfer model receives the power generation rate from the neutron kinetics model and supplies the heat flux distribution to the fuel channel thermal-hydrauhcs model. Using the coolant temperature distribution and heat transfer coefficients from the fuel channel thermal-hydraulic model, it feeds the fuel average temperature distribution back to the neutron kinetics model (Doppler feedback effect). 

The Super LWR has self-controllability of the reactor power against loss of flow and reactivity insertion, like LWRs, due to coolant density and Doppler feedbacks although reverse-flow in the downward-flow water rods slightly complicates the behavior of density feedback. The wide-range sensitivity analyses show that variation of the feedback coefficients does not significantly influence the self-controllability or the safety margin. 

An assumption is made that the reactivity feedback for the reactor is divided evenly between the fuel Doppler effect and feedback from the core structural materials. Doppler temperature feedback is a function of the square root of the volume average temperature of the volume to which the feedback is being applied. The overall temperature feedback is divided between two heat structures the fuel pin structure for Doppler feedback and the core block structure for geometric feedback. Axial feedback coefficients are based on power squared weighting of the Doppler term calculated from the tabular input, and a geometric coefficient assumed to be half of the total temperature feedback. The... 

---