A linear equation of characteristic time profile of power in subcritical quasi-steady state

ABSTRACT

An equation of power in subcritical quasi-steady state has been derived based on one-point kinetics equations for the purpose of utilizing it for the development of timely reactivity estimation from complicated time profile of neutron count rate. It linearly relates power, P, to a new variable q, which is a function of time differential of the power.

It has been confirmed that the points (q, P) calculated by using one-point kinetics code, AGNES, are perfectly in a line described by the new equation and that the points (q, P) calculated from transient subcritical experiment data measured by using TRACY made a line with a slope indicated by the new equation.

KEYWORDS:

• Subcriticality
• reactivity
• measurement
• power profile
• one-point kinetics
• nuclear criticality safety
• reactor kinetics

1. Introduction

There are strong needs for real-time reactivity estimation of an equipment containing fissile materials on detecting abnormal radioactivity. A real-time reactivity monitoring system could avoid inadvertent criticality and reduce the exposure risk of public and operators.

For the development of the estimation method, experimental data are necessary. There are some data measured by using the TRAnsient experiment Criticality facilitY (TRACY), in which core is a cylindrical tank filled with uranyl nitrate solution. In subcritical transient experiments, three types of neutron count rate profile have been recorded under the condition, shown in Table 1 [1]. As shown in Figure 1, at the beginning of each Time Frame, TF, the Pulsed Neutron Source, PNS, was switched-on in TF1, 3, 7, a positive reactivity was instantaneously inserted in TF4, 8, a negative reactivity was instantaneously inserted in TF5 and PNS was switched-off in TF2, 6, 9. PNS produced 14 MeV neutrons by D-T reaction and its reaction rate was 100 per second, which was enough to produce and inject neutrons into the core stably. In each TF, the power gradually approached a stable level during the quasi-steady state which started after the rapid change of power, but before it reached a stable level, next TF started. The reactivity and neutron source strength were constant during each quasi-steady state.

Table 1. Condition of a subcritical transient experiment by using TRACY [1].

Fuel	Uranyl nitrate aqueous solution
^235U enrichment	9.98 wt%
Uranium concentration	401.8 g/L
Acid molarity	0.59 mol/L
Solution height	44.48 cm

Y. Yamane: An equation of characteristic time profile of power in subcritical quasi-steady state.

It is common to use the one-point kinetics equations,

(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1)

(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2)

to simulate power profile, and they are utilized in reactivity estimation under the assumption of the linear relationship between power and neutron count rate. During quasi-steady states, the term $\dot{P}$ of Equation (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) can be neglected because it is much smaller than other terms of the right-hand side, which, however, does not mean $\dot{P}$ is exactly zero. The power gradually changes as the terms $C_i$ change. Inverse kinetics solves the equations with some conditions and methods such as: the values of ${\beta }_{eff}$ and S are given [2], it is near critical state and $S=0$ [3], prompt jump approximation is used to simplify Equation (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) [4], Equation (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) is linearized to eliminate source term [5] and the combination of those methods [6]. Extended Kalman Filtering method [7] uses Equations (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) and (2)(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2) (1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) for transition of the state and improves estimated value by each time step from the initial guess. Integral method [8] uses the time integral of the number of neutrons up to infinity to eliminate the terms of precursors.

Figure 1. Neutron count rate profile measured in a TRACY experiment [1]. Black marker shows neutron count rate at every 0.01 s. Grey line shows an averaged data by using the Savitzky-Golay method. A grey circle in step 6 shows an example of overshooting.

Those methods need a stable level of neutron count rate for the initial guess of variables or to utilize its value to eliminate the terms $C_i$ and S. It is difficult to apply those methods to the TRACY data because there is no stable state observed in each TF. For the timely reactivity estimation in each TF, a simpler equation relating power profile to reactivity would be useful.

The purpose of this study is to find a characteristic equation of power in subcritical quasi-steady state that is expected to be useful for the development of timely reactivity estimation. A differential equation of power has been found and it is much simpler than the original one-point kinetics equations. It is satisfied by the time profile of power in subcritical quasi-steady states observed right after a sudden or rapid change of reactivity and/or the insertion or removal of Neutron Source, NS.

2. Derivation

2.1. Precondition

Equations (1(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) ) and (2(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2) ) are the start point where each variable has the conventional meaning: $P$ is power, $\rho$, reactivity, $\mathrm{\Lambda }\left(s\right)$, prompt neutron generation time, ${\lambda }_i\left(s^{-1}\right)$, decay constant of delayed neutron of ith precursor group, $C_i$, amount of ith precursor, S(W/m³/s), neutron source and the dot over a variable means the time differential of the variable, $\dot{x}=\frac{dx}{dt}$. β_eff is effective delayed neutron fraction, in which value depends on the properties such as the composition, inner structure, and outmost geometry of the fissile material. β_i is delayed neutron fraction of the ith group and ${\beta }_{eff}={\sum }_{i=1}^6{\beta }_i$.

The order of the value of some quantities is important in the next section. In critical state, prompt neutron lifetime, ℓ, is generally in the range of 10⁻³ to 10⁻⁵ s in light water reactors. The value of delayed neutron fraction β is 0.0064 for ²³⁵U, 0.0020 for ²³⁹Pu, and 0.0203 for ²³²Th. While effective delayed neutron fraction β_eff is slightly different from β, its order is generally in the range of 10^−‍2 to 10^−‍3. The value of the ith delayed neutron precursor decay constant, ${\lambda }_i$, for ²³⁵U is shown in Table 2 [9]. The value of those quantities in a subcritical state may be different from one in critical state but such a difference is not so large.

Table 2. Fission yield relative fraction and delayed neutron precursor decay constant of ²³⁵U and ²³⁹Pu [5].

	Fission yield relative fraction bi		Delayed neutron precursor decay constant λi^*
No. of group i	^235U	^239Pu	^235U	^239Pu
1	0.038	0.038	0.0127	0.0129
2	0.213	0.280	0.0317	0.0311
3	0.188	0.216	0.116	0.134
4	0.407	0.328	0.311	0.332
5	0.128	0.103	1.40	1.26
6	0.026	0.035	3.87	3.21

The values are converted by using ${\lambda }_i=\frac{ln2}{T_{1/2i}}$.

Y. Yamane: An equation of characteristic time profile of power in subcritical quasi-steady state.

The power is assumed to be low so that temperature reactivity feedback is ignorable. No other reactivity feedback is considered for simplicity. A positive or negative reactivity is inserted and/or NS is inserted or removed; after a rapid change of power, it becomes a quasi-steady state where $\dot{\rho }=\dot{S}=0$.

2.2. Equation of power in quasi-steady state

Equation (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) and the summation of Equation (2)(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2) give

(3) $\begin{array}{l}\dot{P}=\frac{\rho }{\Lambda }P-\sum _{i=1}^6{\dot{C}}_i+S,\\ \sum _{i=1}^6{\dot{C}}_i=\frac{\rho }{\Lambda }P-\dot{P}+S\cdot \end{array}$(3)

By using the following definition:

(4) $\mu \equiv \frac{{\sum }_{i=1}^6{\lambda }_i{\dot{C}}_i}{{\sum }_{i=1}^6{\dot{C}}_i},$(4)

the time differential of Equation (1)(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) becomes

(5) $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\Lambda \stackrel{\mathrm{..}}{p}=\left(\rho -{\beta }_{eff}\right)\dot{P}+\dot{\rho }P+\Lambda \mu {\displaystyle \sum _{i=1}^6{\dot{C}}_i}+\Lambda \dot{S},\\ =\rho \left(\dot{P}+\mu P+\mu \frac{\Lambda \text{S}}{\rho }\right)+\dot{\rho }P-\left({\beta }_{eff}+\Lambda \mu \right)\dot{P}+\Lambda \dot{S}\end{array}$(5)

From Equation (5)(5) $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\Lambda \stackrel{\mathrm{..}}{p}=\left(\rho -{\beta }_{eff}\right)\dot{P}+\dot{\rho }P+\Lambda \mu {\displaystyle \sum _{i=1}^6{\dot{C}}_i}+\Lambda \dot{S},\\ =\rho \left(\dot{P}+\mu P+\mu \frac{\Lambda \text{S}}{\rho }\right)+\dot{\rho }P-\left({\beta }_{eff}+\Lambda \mu \right)\dot{P}+\Lambda \dot{S}\end{array}$(5) , the reactivity is denoted as

(6) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P+\frac{\Lambda S}{\rho }}{\dot{P}}}$(6)

Because the stable level of power, $P_{\mathrm{\infty }}$, corresponding the reactivity, ρ, satisfies

(7) $\begin{array}{c}{P}_{\mathrm{\infty }}=\frac{\mathrm{\Lambda }S}{-\rho },\end{array}$(7)

Equation (6)(6) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P+\frac{\Lambda S}{\rho }}{\dot{P}}}$(6) becomes

(8) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P-P_{\infty }}{\dot{P}}}$(8)

The smallness of the second and third terms of the numerator in Equation (8)(8) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P-P_{\infty }}{\dot{P}}}$(8) is as follows: $\dot{S}/\dot{P}$ must be zero because $\dot{S}=0$ during quasi-steady state while $\dot{P}\ne 0.$ The smallness of $\ddot{P}/\dot{P}$ is not trivial. For a function of time, $f\left(t\right)=1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\lambda t\right)$, with constant λ, for an example, it is easy to find that $\ddot{f}/\dot{f}=-\lambda \ne 0$.

It is well known that the in-hour equation, which is equivalent to one-point kinetics equations, has a solution with seven components corresponding to Ʌ and λ_i for a power profile. The general solution of its homogeneous form is shown in the reference [10], from which the general solution with an external neutron source in each time frame shown in Figure 1 is easily derived as $P=P_{\mathrm{\infty }}-{\sum }_{i=0}^7{A}_i\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\gamma }_{i}t\right)$ with constant A_i and γ_i. The initial rapid change in the power profile is made by the decay of the term of large γ_i. In the period after such rapid change, the term of the smallest γ₁ becomes dominant and the solution comes close to the following equation:

(9) $\begin{array}{c}P=P_{\mathrm{\infty }}-A_{1}exp\left(-{\gamma }_{1}t\right)\end{array}$(9)

where 0 < γ₁ < λ₁ and A₁ is a constant. λ₁ is, for an example, 0.0127 for ²³⁵U. From consideration in the previous paragraph, the following equation is satisfied after rapid change in the power:

(10) $\left|\frac{\stackrel{\mathrm{..}}{P}}{\dot{P}}\right|={\gamma }_1<{\lambda }_1={10}^{-2}{s}^{-1\cdot }$(10)

This condition is a sufficient condition and not a necessary condition because what is needed is $\ddot{p}/\dot{p}\leqq \mu$ and it is shown in the next section that $\mu \leqq 3\sim 4s^{-1}$. From these, the order of the second term in the numerator of Equation (8)(8) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P-P_{\infty }}{\dot{P}}}$(8) is less than O(10⁻⁵), while the order of β_eff is O(10⁻³). In conclusion, with the condition $\dot{\rho }=\dot{S}=0,$ the second and third terms are negligibly small:

(11) ${\beta }_{eff}\gg \Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{P}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}},$(11)

and Equation (8)(8) $\rho =\frac{{\beta }_{eff}+\Lambda \left(\mu +\frac{\stackrel{\mathrm{..}}{p}}{\dot{P}}-\frac{\dot{S}}{\dot{P}}\right)-\dot{\rho }\frac{P}{\dot{P}}}{1+\mu \frac{P-P_{\infty }}{\dot{P}}}$(8) becomes

(12) $\begin{array}{c}{\rho }_{\mathrm{\$}}=\frac{1}{1+\mu \frac{P-P_{\mathrm{\infty }}}{\dot{P}}},\end{array}$(12)

where ρ_$ = ρ/β_eff.

With the following definitions:

(13) $\begin{array}{c}{\alpha }_y\equiv \frac{1}{{\rho }_{\mathrm{\$}}}-1,\end{array}$(13)

(14) $\begin{array}{c}q\equiv \frac{\dot{P}}{\mu },\end{array}$(14)

We finally have the target equation:

(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15)

Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) looks the equation of a line in q-P plane with constant slope, α_y, and intercept, P_∞. The linearity of points (q, P) is a very useful nature because if points (q, P) are known, α_y and P_∞ are able to be estimated by linear fitting.

It must be noted that Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) is not valid when the power is rapidly changing. In a short period right after the change of the condition, one-point kinetics may not describe the power profile well due to the existence of the higher mode flux.

It is worth to note that in Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) the information of the geometrical property such as the inner structure and outmost geometry is contained only in α_y through the reactivity ρ_$ and only μ contains b_i and λ_i which are unique to each fissile material.

2.3. Ratio μ

The dependency of μ on the other parameters was investigated to understand the nature of Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) . By using a function f_i(t) defined as

(16) $\begin{array}{c}{f}_i\left(t\right)\equiv \frac{\mathrm{\Lambda }}{{\beta }_i}{C}_i\left(t\right),\end{array}$(16)

integral of Equation (2)(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2) gives

(17) $\begin{array}{c}{f}_i\left(t\right)=\underset{-\mathrm{\infty }}{\overset{t}{\int }}P\left(t\right)e^{{\lambda }_i\left(\tau -t\right)}d\tau ,\end{array}$(17)

(18) $\begin{array}{c}{\dot{f}}_i\left(t\right)=\mathrm{P}\left(t\right)-{\lambda }_i{f}_i\left(t\right),\end{array}$(18)

and Equation (4)(4) $\mu \equiv \frac{{\sum }_{i=1}^6{\lambda }_i{\dot{C}}_i}{{\sum }_{i=1}^6{\dot{C}}_i},$(4) becomes

(19) $\mu \equiv \frac{{\sum }_{i=1}^6{\lambda }_i{b}_i{f}_i}{{\sum }_{i=1}^6{b}_i{f}_i}$(19)

where b_i is the ith delayed neutron relative yield, $b_i={\beta }_i/{\beta }_{eff}$, for uranium 235 is shown, for an example, in Table 2. Equation (19)(19) $\mu \equiv \frac{{\sum }_{i=1}^6{\lambda }_i{b}_i{f}_i}{{\sum }_{i=1}^6{b}_i{f}_i}$(19) indicates that $\mu \leqq {\lambda }_6=3\sim 4s^{-1}$ and μ goes to λ₁ as time goes to infinity because each ḟ_i, as the same as Ċ_i, goes to zero with each decay constant λ_i, and the term with minimum λ_i finally remains. It is clear that μ depends only on b_i, λ_i, P. Therefore, the essential parameters of Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) are b_i and λ_i, which are unique for each fissile isotope and not dependent on the geometrical property of the fissile material. That also means the power profile in quasi-steady states does not depend on the effective delayed neutron fraction, β_eff, or mean neutron generation time, Ʌ.

3. Verification by one-point kinetics calculation

The TRACY experiment was simulated by using one-point kinetics code, AGNES [11], which explicitly numerically solves Equations (1(1) $\begin{array}{c}\dot{P}=\frac{\rho -{\beta }_{eff}}{\mathrm{\Lambda }}P+\sum _{i=1}^6{\lambda }_i{C}_i+S\end{array}$(1) ) and (2(2) $\begin{array}{c}{\dot{C}}_i=\frac{{\beta }_i}{\mathrm{\Lambda }}P-{\lambda }_i{C}_i,\end{array}$(2) ), and simulate power profiles of transient experiment very well [8]. The TRACY experiment is simulated with the parameter values in Table 3. Because relative power profile was concerned here, the value of neutron source strength was arbitrarily chosen and set as 10⁻² W/m³/s. Reactivity feedbacks of temperature and void were taken as zero because the power was very low under subcritical condition. The power profiles similar to Figure 1 are obtained as shown in Figure 2. $\mu$ was calculated by using Equations (17)(17) $\begin{array}{c}{f}_i\left(t\right)=\underset{-\mathrm{\infty }}{\overset{t}{\int }}P\left(t\right)e^{{\lambda }_i\left(\tau -t\right)}d\tau ,\end{array}$(17) , (18),(18) $\begin{array}{c}{\dot{f}}_i\left(t\right)=\mathrm{P}\left(t\right)-{\lambda }_i{f}_i\left(t\right),\end{array}$(18) and (19(19) $\mu \equiv \frac{{\sum }_{i=1}^6{\lambda }_i{b}_i{f}_i}{{\sum }_{i=1}^6{b}_i{f}_i}$(19) ), and variable q(t) of Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) was also calculated. The calculated points (q, P) are plotted in Figure 3. They are perfectly in a line described by Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) . At the beginning of the quasi-steady state in each TF, the points (q, P) were at the end far from the vertical axis, $q=0$. During the quasi-steady state, the points (q, P) moved towards the vertical axis and made a line, which is completely the same as one described by Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) . It is clearly seen in TF1, 3, 4, 8 that the points (q, P) did not reach the vertical axis because the next TF started before the power reached a stable level.

Figure 2. Simulation of the TRACY experiment by using one-point kinetics code AGNES. Grey line shows the calculated power. Black markers show the data at every 0.01 s, which were used in the analysis shown in Figure 3.

Table 3. Kinetics parameters used in one-point kinetics calculation

Parameter		Value
Prompt neutron generation time Λ (s)		4.6E-5
Effective delayed neutron fraction βeff		7.5E-3
Fission yield fraction
	β1	2.50841E-04
	β2	1.62669E-03
	β3	1.46869E-03
	β4	2.95682E-03
	β5	8.77736E-04
	β6	3.19219E-04
Delayed neutron precursor decay constant (1/s)
	λ1	1.27032E-02
	λ2	3.17042E-02
	λ3	1.15272E-01
	λ4	3.11672E-01
	λ5	1.40032E+00
	λ6	3.87437E+00

Y. Yamane: An equation of characteristic time profile of power in subcritical quasi-steady state.

4. Application to TRACY experiment data

Neutron count rate is used as the power in Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) following conventional way for the reactivity estimation based on one-point kinetics. Savitzky-Golay method [12] was applied to the TRACY experiment data with 0.01 s sampling time to calculate the time profiles of average neutron count rate and its time differential. An averaged point was calculated from the original data for the period of 10 s by applying SG method. Figure 1 shows the averaged profiles and the original data.

An extra Moving Average, MA, was applied to $\dot{P}$ data calculated by using SG method in order to reduce the oscillation originated from SG method for clear visibility. The period of MA was less than 10 s to avoid false linearity due to over-averaging.

As shown in Figure 4, the calculated points (q, P) made a line with the slope, α_y, calculated by using Equation (13)(13) $\begin{array}{c}{\alpha }_y\equiv \frac{1}{{\rho }_{\mathrm{\$}}}-1,\end{array}$(13) , which is explained in detail below. The length of the lines was shorter than the lines in Figure 3 because applying the SG method and MA distorted the data at the both ends and those distorted data were not plotted. For an example, in Figure 1 step 6, a circle indicates an overshooting due to the following rapid change of power. In Figure 4, the points (q, P) calculated from such distorted data were not plotted.

Figure 3. Points (q, P) calculated by using one-point kinetics code AGNES. Black marker shows the points (q, P) at every 0.01 s. Grey line shows a straight line with specified slope. Black-dotted line shows the vertical axis.

Figure 4. Points (q, P) calculated from TRACY experiment data. Black or Grey marker shows the points (q, P). Grey line shows a straight line with specified slope.

Figure 4(a) shows the result of −1.4$ cases. TF1 was source insertion case and TF4 and 8 were positive reactivity insertion. In all cases, the points (q, P) made a line with slope −1.7 corresponding to −1.4$ according to Equation (13)(13) $\begin{array}{c}{\alpha }_y\equiv \frac{1}{{\rho }_{\mathrm{\$}}}-1,\end{array}$(13) . The value of the intercept P_∞, of the grey lines for TF1, 4, 8 was 6.4 × 10³, 6.5 × 10³, 6.6 × 10³, respectively, and their difference is up to 3%.

Figure 4(b) shows the result of PNS-off, similar to source jerk, cases. In the cases of –‍1.4$, TF2 and 9, the points (q, P) made a line with slope −1.7 and in the case of −3.1$, TF6, the points (q, P) made a line with slope −1.3 corresponding to −3.1$. In each case, the points (q, P) moved toward the origin, the point of zero power, along the line indicated by Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) .

Figure 4(c) shows the result of negative reactivity insertion case, TF5, from −1.4$ to –‍3.1$. The points (q, P) made a line with slope −1.3.

Cases TF3 and 7 were not considered because PNS took several ten seconds to start stable release of neutrons, which very much distorted the initial power profile for accurate analysis. In case TF1, the latter half of the data were used for the analysis.

5. Discussion

In Figure 4, the points (q, P) were calculated by using MA for better visibility. But using MA causes worse accuracy in reactivity estimation by distorting the profile of data and decreasing the number of useful data. Therefore, a good procedure with proper averaging must be developed for useful reactivity estimation. Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) is also expected to be useful in determining the value of P_∞ and S in each reactivity level, and its nature should be investigated for more application.

In practical use, positive reactivity insertion is the most important case and two important examples in which a quasi-steady state could appear are as follows:

1. Plutonium solution was inadvertently flown into a waste storage tank, neutron count rate increased beyond the preset level and the flow was terminated.

2. In Fukushima Daiichi Nuclear Power Plant, during retrieval operation of fuel debris, neutron count rate increased beyond the preset level and the operation was terminated.

In each case, after the termination of the flow or operation, the power is expected to satisfy the equation of power in subcritical quasi-steady state (Equation (15)(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) )(15) $\begin{array}{c}P={\alpha }_{y}q+P_{\mathrm{\infty }}.\end{array}$(15) .

6. Conclusion

A new simple linear equation of power in subcritical quasi-steady state has been derived based on one-point kinetics equations. It shows that after the stop of reactivity change and/or neutron source strength change, the power profile depends on only ρ_$, b_i and ℓ_i. The numerical calculation by solving one-point kinetics equations showed that the points (q, P) are perfectly in a line described by the equation. The application of the equation to TRACY experiment data showed that points (q, P) made a line with the slope indicated by the equation.

That implies the possibility of new timely reactivity estimation from complicated power profile, in which required parameters are only b_i and ℓ_i. Further investigation of the nature of the equation is expected to benefit the criticality safety of practical situations such as fuel debris retrieval work, and so on.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Disclosure Statement

No potential conflict of interest was reported by the author.