Corresponding author: Kirill S. Kupriyanov ( bdf1@mail.ru ) Academic editor: Georgy Tikhomirov
© 2021 Kirill S. Kupriyanov, Vladimir V. Pereverzentsev.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Kupriyanov KS, Pereverzentsev VV (2021) Analytical model for determining the leakage albedo component for a direct cylindrical channel passing through the nuclear reactor protective layer. Nuclear Energy and Technology 7(2): 9195. https://doi.org/10.3897/nucet.7.68941

The task of determining the radiation situation, including neutron and gammaquantum flux density, radiation spectrum, specific volumetric activity of radioactive gases in the air, etc. behind the protective composition having inhomogeneities, has always been important in matters of radiation safety. One of the ways to solve the problem of determining gamma radiation fluxes was to divide the total ionizing radiation flux into four components: lineofsight (LOS), leakage, lineofsight albedo, and leakage albedo, and obtain an analytical solution for each component. The first three components have been studied in detail in relation to simple geometries and there are analytical solutions for them, but there is no such a solution for the last component. The authors of this work have derived an analytical representation for the leakage albedo component, which, in contrast to numerical methods (such as Monte Carlo methods), makes it possible to analyze the effect of inhomogeneities in protective compositions on the radiation environment as well as to quickly obtain estimated values of fluxes and dose rates. Performing a componentbycomponent comparison, it becomes possible to single out the most significant mechanisms of the dose load formation behind the nuclear reactor protection, to draw conclusions about the effectiveness of design solutions in the protection design and to improve the protection at significantly lower computational costs.
Finally, the authors present calculations for the four components of the total ionizing radiation flux for various parameters of the cylindrical inhomogeneity in the reactor protection. Based on the obtained values, conclusions are made about the importance of taking into account the leakage albedo component in the formation of the radiation situation behind the core vessel.
Leakage albedo, inhomogeneities in protective compositions, radiation protection.
Inhomogeneities in protective compositions are subdivided into elementary (simple) and complex ones. Elementary inhomogeneities include those in which the radiation field does not depend on the field of a neighboring inhomogeneity. The study of complex inhomogeneities is a more general problem and, as a rule, does not have an analytical solution (
The authors consider an elementary straight cylindrical channel with a diameter of 2a passing through the protective layer. On the one hand, there is a plane uniform power source N_{0}; the detection point D is at the outlet from the protection on the channel axis (Fig.
In the figure, the ray shows the formation of the leakage albedo component — the ray leaves the zone outside the channel, goes into the channel region, being physically attenuated, is reflected from the channel wall and hits the detection point D. Let us obtain a solution for this component of the flux. The general law of physical attenuation for ionizing radiation is written as exp(–m_{0}L) (
• m_{0} º m – for gamma quanta ;
• m_{0} º [l(E_{n} > E_{n}^{*})]^{–1} – for fast neutrons;
• m_{0} º Sa – for thermal neutrons.
For simplicity, we shall first consider the plane problem, and then move on to the spatial solution. Let us consider a top view (from the end of the channel) and mark the elements responsible for the formation of the leakage component (Fig.
Let us consider the right section: at radius R, we select the area dS to the right of the tangent. The incident flux is attenuated over the entire section of length l_{1}. Since a is the angle between the ray drawn from the center of the circle through the point P and the ray drawn from the center of the circle to the center of the area dS, then for a Î [0, a_{bd}] we can write the expression for the flux:
$d\Phi =R\frac{{N}_{0}f\left(\theta \right)}{{{l}_{1}}^{2}}dRd\alpha \xb7\mathrm{exp}\left({\mu}_{0}{l}_{1}\right)$ (1)
where f (q) is the angular distribution of the source radiation; l_{1} = (R^{2} + a^{2} – 2Ra×cos(a))^{1/2} (by the cosine theorem); a_{bd} = arccos(a/R).
Assuming that the source is isotropic (f (q) = 1/(2p)), and integrating, we obtain an expression for the flux incident on the channel wall from the part of the ring of thickness dR on the right side (the expression is multiplied by 2, since the flow is summed up and down from the point P):
$d{\Phi}_{1}=2{N}_{0}{\int}_{0}^{{\alpha}_{bd}}\frac{R}{2\pi {l}_{1}^{2}}dRd\alpha \xb7\mathrm{exp}\left({\mu}_{0}{l}_{1}\right)$ (2)
Similarly, we can obtain the total incident flux from the part of the ring on the left side. The difference between the right and left sides along the length of the attenuation section (see Fig.
$d{\Phi}_{2}=2{N}_{0}{\int}_{{\alpha}_{\mathrm{bd}}}^{\pi}\frac{R}{2\pi {l}_{2}^{2}}dRd\alpha \xb7\mathrm{exp}\left({\mu}_{0}\left[{l}_{1}{l}^{\u2019}\right]\right)$ (3)
where l_{2} = l_{1} = (R^{2} + a^{2} – 2Ra×cos(a))^{1/2} (by the cosine theorem); l¢ = 2a[1 – sin^{2}(a)×(R/l_{2})^{2}]^{1/2}. The expression for l¢ is obtained from the relations h¢/R = – a×sin(a)/l^{2} (the sine theorem for a triangle with side l_{2}) and l¢/2 = (a^{2} – (h¢)^{2})^{1/2}. Further on, equal lengths l_{2} and l_{1} will be denoted by the symbol l.
Now, we shall turn to the spatial problem (Fig.
To illustrate how the new geometric attenuation lengths are calculated, Fig.
The length of geometric attenuation L is determined by the expression (l^{2} + z^{2})^{1/2}; whereas the expression [(z (l – l¢)/l)^{2} + (l – l¢)^{2}]^{1/2} is used to determine the length of physical attenuation.
Then formulas (2) and (3) can be represented as
$d{\Phi}_{1}=2{N}_{0}{\int}_{0}^{{\alpha}_{\mathrm{bd}}}\frac{R}{2\pi \left({l}^{2}+{z}^{2}\right)}dRd\alpha \xb7\mathrm{exp}\left({\mu}_{0}\sqrt{\left({l}^{2}+{z}^{2}\right)}\right)$ (4)
$d{\Phi}_{2}=2{N}_{0}{\int}_{{\alpha}_{bd}}^{\pi}\frac{R}{2\pi \left({l}^{2}+{z}^{2}\right)}dRd\alpha \xb7\mathrm{exp}\left\{{\mu}_{0}\sqrt{{\left[z\left(l{l}^{\text{'}}\right)/l\right]}^{2}+{\left(l{l}^{\text{'}}\right)}^{2}}\right\}$ (5)
In most practical cases, we can consider the source as an infinite plane and perform integration over the radius with an upper limit equal to infinity. Thus, the total flux incident at the point P from the infinite plane is
${\Phi}_{tot}={\int}_{a}^{\infty}\left(d{\Phi}_{1}+d{\Phi}_{2}\right)dr$
To obtain the values of the reflected flux, we shall use the value of the numerical differential albedo: this value depends on the angle of incidence q, the angle of reflection y and the energy of the ionizing radiation flux. The angle of incidence depends on the angle a, therefore, this value must be taken into account even before the first integration over the angle a.
The task is axisymmetric; therefore, integrating the differential of the flux incident onto the lateral surface along a narrow ring, we finally obtain a solution for the leakage albedo component as the sum of two components:
F_{leak.alb} = F_{leak.alb1} + F_{leak.alb2};
$\begin{array}{l}{\Phi}_{\text{leak.alb}1}={\int}_{0}^{h}{\int}_{a}^{\infty}{\int}_{{\alpha}_{\mathrm{bd}}}^{\pi}\frac{{N}_{0}R}{\pi \left({l}^{2}+{z}^{2}\right)}dRd\alpha \times \\ \times \mathrm{exp}\left({\mu}_{0}\sqrt{{l}^{2}+{z}^{2}}\right)\xb7\frac{{a}_{n}\left({E}_{0},{\theta}_{0},\psi \right)}{(hz{)}^{2}+{a}^{2}}2\pi \alpha dz\end{array}$
$\begin{array}{l}{\Phi}_{\text{leak.alb2}}={\int}_{0}^{h}{\int}_{a}^{\infty}{\int}_{0}^{{\alpha}_{\mathrm{bd}}}\frac{{N}_{0}R}{\pi \left({l}^{2}+{z}^{2}\right)}dRd\alpha \times \\ \times \mathrm{exp}\left\{{\mu}_{0}\sqrt{{\left[z\left(l{l}^{\u2019}\right)/l\right]}^{2}+{\left(l{l}^{\u2019}\right)}^{2}}\right\}\xb7\frac{{a}_{n}\left({E}_{0},{\theta}_{0},\psi \right)}{(hz{)}^{2}+{a}^{2}}2\pi adz\end{array}$
Using the Heaviside function H(a – a_{bd}), we can write the solution in a single integral:
$\begin{array}{c}{\Phi}_{\text{leak.alb}}={\int}_{0}^{n}{\int}_{a}^{\infty}{\int}_{0}^{\pi}\frac{{N}_{0}R}{\pi \left({l}^{2}+{z}^{2}\right)}dRd\alpha \times \\ \times \frac{\mathrm{exp}\left\{{\mu}_{0}\left[l{l}^{\u2019}H\left(\alpha {\alpha}_{\mathrm{bd}}\right)\right]\sqrt{(z/l{)}^{2}+1}\right\}\xb7{a}_{n}\left({E}_{0},{\theta}_{0},\psi \right)}{(hz{)}^{2}+{a}^{2}}2\pi adz\end{array}$ (6)
To check the obtained formula, Monte Carlo calculations were performed using SERPENT (a multipurpose threedimensional Monte Carlo particle transport code) (
In the model calculation, aluminum was used as a protective material, and the source was set to be monoenergetic with energy of gamma quanta equal to 1.25 MeV. To obtain a flux of gamma quanta on the channel axis at the outlet from the protection, a small finite volume was set in the model, sufficient to register gamma quanta emitted from the source. The results of model calculations, as well as calculations by analytical formulas, are presented in Tab.
a, cm  h, cm  SF_{Serpent}, cm^{–2}s^{–1}  SF_{analyt.}, cm^{–2}s^{–1}  e = [(SF_{analyt.} – SF_{Serpent})/ SF_{Serpent}] ×100, % 

15  135  0.011  0.01  –9.1 
15  270  3.501×10^{–3}  2.291×10^{–3}  –34.6 
30  135  0.037  0.036  –2.7 
30  405  5.252×10^{–3}  3.875×10^{–3}  –26.2 
10  200  2.241×10^{–3}  1.968×10^{–3}  –12.2 
Let us now analyze the results obtained.
• Comparing the fluxes obtained analytically and by the Monte Carlo method, one can see that the analytical result always gives lower values.
• Minimum errors are obtained in the case when the inhomogeneity is large and the main contribution to the flux formation is made by the lineofsight component. If the influence of the leakage components grows, the error increases, which is caused by the violation of the correctness of the assumptions made in the derivation of the analytical formulas.
Let us estimate the contribution of the leakage albedo component to the total ionizing radiation flux density at the detection point. Calculations for a particular case will be presented below.
The monoenergetic ionizing radiation source is ^{60}Co; the reflective surface is aluminum. The values of a_{n} (E_{0}, q_{0}, y) were taken from (
Calculations of individual components of the ionizing radiation flux density
a, cm  h, cm  m_{0}, cm^{–1}  F_{LOS}, cm^{–2}s^{–1}  F_{leak.}, cm^{–2}s^{–1}  F_{LOS alb.}, cm^{–2}s^{–1}  F_{leak. alb.}, cm^{–2}s^{–1}  F_{leak. alb.} SF 
15  350  0.3  9.18×10^{–4}  3.93×10^{–6}  1.786×10^{–4}  7.451×10^{–5}  0.063 
5  350  0.3  1.02×10^{–4}  1.087×10^{–9}  6.795×10^{–6}  1.462×10^{–5}  0.118 
5  115  0.3  9.443×10^{–4}  5.759×10^{–5}  1.856×10^{–4}  2.44×10^{–4}  0.17 
30  350  0.3  3.66×10^{–3}  5.341×10^{–5}  1.17×10^{–3}  2.043×10^{–4}  0.04 
15  115  0.3  8.435×10^{–3}  5.265×10^{–4}  3.104×10^{–3}  9.932×10^{–4}  0.076 
15  115  3  8.435×10^{–3}  8.383×10^{–6}  3.104×10^{–3}  9.452×10^{–5}  0.0081 
5  350  3  1.02×10^{–4}  0  6.795×10^{–6}  1.377×10^{–6}  0.012 
Let us analyze the results presented in this table:
– For substances with high m_{0} the contribution of the leakage components is orders of magnitude less than that of the lineofsight components.
– As height h increases, the contribution of the leakage albedo can both decrease and increase.
– With a decreasing in size, the value of the lineofsight flux decreases more intensively than the value of the leakage albedo flux, which increases the contribution of the latter to the total flux.
Therefore, it is important to take into account the leakage albedo in the case of small channels in protections with ‘low’ values of m_{0}. For a particular case from the given example, the leakage albedo can be 17% of the total flux.
Despite the rapid development of numerical methods in the calculation of the radiation environment, analytical solutions still find their application in the initial estimates of radiation fields, in the study of the dependences of the obtained fluxes of ionizing radiation, as well as for verification of software systems (