Inhour equation

The asymptotic period is one of the most commonly used methods for reactivity determination, whereby the measured asymptotic period is related to reactivity via the Inhour equation (7, 8). The definition of reactivity inferred from the Inhour equation depends on the definition of the reactor s effective kinetic parameters. 

The Inhour equation relating the asymptotic period to the static reactivity is... 

We shall refer to as the asymptotic-period reactivity. The Inhour equation for this reactivity can be obtained by multiplying Eq. (19) by 0q, integrating over phase space and rearranging ... 

The calculation of the effective kinetic parameters of the Inhour equation for the asymptotic-period reactivity requires only one distribution 0, for each subcritical configuration. The effective kinetic parameters for the... 

Inhour equation for the static reactivity [Eq. (13)] are functions of two such distributions, (f), and 4>. The asymptotic-period reactivity is, therefore, the natural reactivity to infer from the Inhour equation. 

An additional equation can be derived from (9) and (10) which is very important in experimental low power reactor physics. When a reactor undergoes a step-insertion of reactivity, which renders the reactor slightly supercritical, for example, and when transient disturbances die out, the neutron and delayed emitter populations will behave as e, where o) is the largest root of the inhour equation, obtained from (9) and (10) by taking fi(t) F(t) e ... 

The inhour equation for both approaches takes the form... 

Figure 2 shows the results of sample calculations utilizing the homogenized and two-region inhour equations. The examples are infinite slab reactors with uranium-233 as the core and uranium-235 as the blanket fuels. The choice of placing uranium-233 on the inside is because of a particular study under way at the time the calculations were made. Fuel concentrations... 

The problem of immediate concern is that of the calibration of a control rod in a thermal reactor. By control rod calibration is meant the determination of the reactivity worth of a control rod per unit distance of travel, usually in a vertical direction. Such a calibration, particularly in experimental critical facilities, is of great importance in connection with a variety of reactor experiments. It is rather easy to calibrate a control rod, particularly in a low power critical assembly. A rod is withdrawn a few centimeters in a just critical reactor. As soon as the flux takes off on an exponential ascent the reactor period can be measured with the aid of appropriate detectors and recording devices. A relationship between the reactor period and reactivity is afforded from the inhour equation. Thus, the reactivity worth of a control rod per unit length of travel in a certain area of the reactor is determined. The rod may be calibrated over its entire length of travel by the expedient of maintaining criticality by the opposing... 

The blade being calibrated will then be moved an amount h, and the resulting reactivity, = , will be determined from the resulting period. Following this measurement, a new height h2 will be chosen and the process repeated. To determine the reactivity associated with the element movement h, it is necessary to use the inhour equation. 

The foregoing tells us that to determine the reactivity associated with the blade movement h, the resultant asymptotic period of the reactor must be measured. can then be calculated from the inhour equation. Point 2 says that after the withdrawal of the element, the power rise will not be a simple exponential and that there will be a waiting period while the transient terms did out. 

Supercriticality definition of critical, prompt critical, reactivity, the inhour equation approximations to the inhour equation for long periods and very short periods reactivity units--preferred % , but mention pern, dollars and inhours, and conversion from one to another negative periods and limitations. 

The solution of the reactor kinetic equations for these conditions yields the INHOUR EQUATION. 

The inhour equation is used to determine reactivity - period relations. Since it is a complicated equation, let s look at some useful approximations to the inhour equation. 

Calculate, using the two approximate forms of the inhour equation, the resulting period for the following reactivity insertions ... 

The Inhour Equation relates reactivity insertion, p, to reactor period, T. Some positive reactivity is inserted into Reactor A, while the same amount of negative reactivity is inserted into Reactor B. After all transients have decayed away, the absolute value of the period will be ... 

The resulting period must be the asymptotic pile period t as designated by the inhour equation, so that a must be identical with l/. a and b are coupled with the coefficients of h and n (Ri and R2) as follows ... 

The ROS is intended to simulate some of the most important and simple cases of reactor kinetics qualitaUvely. It is necessaiy to solve inhour equation to get reactor period Then you can choose control rod diqiosition The aim here is demonstratimi of some features ooncemed with kinetics aspect, so that the reactor IS alw supposed to be at critical state and other parameters such as the core height, number of fiid elements, core radius and so on may be neglected. Siqipose t one rod is placed at the bottom and the other is withdrawn all the way whoe cme of these rixls is to be calibrated. Total worth id the rod depends on its position, radius (r) a matoial (A)... 

From the plot of the inhour equations relating the stable reactor period to kex> ex 0.10% Ak/k. 

Then, by imposing initial conditions, we can eliminate the delayed-neutron-precursor density from the modified diffusion equation. Finally, we simplify the modified diffusion equation and arrange it in the standard "characteristic inhour equation" form, which relates our original function of reactivity to some of the nuclear properties of the materials present in the reactor. 

In this experiment the basic concepts of control rod calibration of a nuclear reactor are considered. Reactivity changes are measured by observing positive asymptotic periods of the reactor. These asymptotic periods are related to reactivity by the reactor kinetic equations and the usual inhour equation. Several examples of kinetic behavior in response to changes in reactivity are qualitatively observed so that operating skill and safe attitudes pertaining to reactor operation may be developed. 

The fine control rod of the AGN-201 reactor will be calibrated in this experiment. The technique used to measure reactivity changes at different rod positions in the core will be measurements of positive asymptotic periods. The reactivity of the assembly corresponding to a measured stable period will be determined from the basic inhour equation, which is derived below. 

The derivation of the inhour equation, which relates period measure-vts to reactivity, involves the use of the basic diffusion equation ... 

The quantity odq is the reciprocal of the stable asymptotic period T. If a measurement of this stable period is made when a reactor is supercritical, the reactivity corresponding to the period can be determined from Eq. (10). Substituting CJO = i/t into the basic inhour equation gives... 

The basic inhour equation derived from the time-dependent diffusion equation is... 

This experiment is designed to acquaint the participant with the use of the danger coefficient method to measure the absorption cross sections of certain elements. The experiment consists of a measurement of the sensitivity of the AGN-201 reactor at the core center with a standard l/v-absorber and subsequent cross-section measurements of a selected group of materials in terms of the reactor sensitivity. Reactivity will be determined by measurement of positive periods with and without the material in the reactor core. The excess reactivity of the supercritical reactor is related directly to the positive period measurement through the basic inhour equation. The results of the experiment will be compared with the known thermal cross sections and a complete analysis of any discrepancies will be made. 

The reactivity changes that occur in the reactor when an absorber is placed at its center will be measured in terms of asymptotic positive periods. The stable period of a supercritical reactor is related to the reactivity through the basic inhour equation. This expression involves the prompt neutron lifetime and the delayed neutron parameters (see Appendix B). 

The reactor is once again brought to a subcritical position and boron sample B is inserted into the glory hole. The procedure for measuring the period is then the same as it was for the Lucite blank and boron sample A. The reactivity of the reactor for each run can be evaluated from the basic inhour equation, which relates the reactor period to the reactivity. A plot of this equation can be found in Appendix B. The reactivity worths of the boron samples is determined by subtracting the... 

The excess reactivity for each run is determined from the plot of the basic inhour equation. The reactivity worth of each sample is found by subtracting the reactivity of the sample run from that of the blank Lucite. Two blank runs are made to ensure that no changes have occurred which might effect the core excess reactivity. The control rods for each of the six runs must of course be at the same positions. Once the worth of each sample is determined, its cross section can easily be calculated knowing the amount of material present in the sample and the sensitivity of the reactor. 

---