Behavior of Tritium Released to the Environment

Abstract

Tritium release from a fusion reactor consists of chronic one during the normal operation and accidental one. Once tritium (T) is released from the reactor, it disperses into environment. Because of its low energy of the β-electron released at T decay, internal exposure by T uptake (inhalation and ingestion) and its biological effects are major concerns. Therefore, it is quite important to estimate and/or understand how dispersed T is finally uptaken in and impact human living in surrounding area. In this chapter, numerical models developed to estimate concentration in the environmental elements, and their distribution of T released from a fusion plant are described. The analysis of T behavior in the environment starts from determining source terms, then follow analyses of spatial transportation by advection and diffusion and migration among environmental elements (compartments), and estimates uptake into human body and finally evaluates dose.

Appendices

Appendix: Analysis of Tritium Behavior in the Atmosphere

The analysis introduced in the main text for T release , transportation, migration, uptake into lives, and so on in surrounding area of a fusion power plant is based on Gaussian puff and plume models which have been well developed for analyzing transportation of air pollutant in the atmosphere . Here, the models are explained in a little detail with modifications of them to analyze the T transportation. Details of the T transportation analysis are in the main text.

Transportation by Advection and Diffusion

Dispersion of T released into the vicinity atmosphere is affected by the meteorological conditions such as wind velocity and solar irradiation, which vary from hour to hour. Therefore, it is hard to analyze and predict the dispersion behavior accurately.

In liquid, gas, or solid that is in static condition, solutes (atoms and molecules) in them diffuse driven by molecular motion, which can be solved analytically by applying Fick’s law with their diffusion coefficients . On the other hand, in the case of dispersion in atmosphere, the solute is carried by air which moves intricately and far faster than molecular diffusion. This is called “turbulent diffusion” and is multiscale phenomenon ranging from molecular motion (~10⁻⁷ m) to global scale (~10⁷ m). Therefore, it is quite hard to analyze completely and to trace all atmospheric molecules in the analyzing area. However, if focusing attention on limited time and spatial scales, the phenomena of the interested region can be analyzed by using simple functions or coefficients as representatives of other scaling phenomena. A lot of analyzing models have been developed for many kinds of scales in time and space; we should select the appropriate one depending on the interest. To analyze T behavior released from a fusion plant during normal operation, for example, the scales might be orders of ~10 years and ~100 km.

Introduction of Traditional Tritium Dispersion Models and Codes

A Gaussian dispersion model is very simple and therefore has been used for the prediction of migration of air pollutant such as industrial fumes. It has been modified for analyzing transportation of radioactive materials in both chronic and accidental releases from a nuclear fission power plant. In particular, a Gaussian plume model is adopted to estimate dispersion of radioactive materials released from a fission power plant in the “Meteorological Guide for Safety Analysis of Nuclear Power Reactor Facilities” [12] which guides how to assess a plan for construction of a nuclear fission power plant in Japan.

• (a) Gaussian puff model

In the Gaussian puff model , the pollutant matter is assumed to be released instantaneously at a time and to expand into a flat spherical puff, the center of which is moving along the wind. The spatial distribution of the matter in the puff is assumed to form the normal (Gaussian) distribution, which satisfies the diffusion equation by Fick’s law. This model is usually applied to the case of the instantaneous release of the pollutant matter such as accidental release . Furthermore, this is the base of subsequently explained plume models to analyze chronic release of pollutants.

Calm condition

Consider that pollutant with the total amount of q is instantaneously released at time, t = 0, from a point (x ₀, y ₀, z ₀) in flat calm. The pollutant expands with time following the Gaussian distribution. Therefore, the concentration of the pollutant in air, c, at an observation point (x, y, z) at time t is expressed as:

$$c\,(x, y, z, t) = \frac{q}{{(2\pi )^{{\frac{3}{2}}} \sigma_{x} \sigma_{y} \sigma_{z} }}{ \exp }\left( { - \frac{{(x - x_{0} )^{2} }}{{2\sigma_{x}^{2} }}} \right){ \exp }\left( { - \frac{{(y - y_{0} )^{2} }}{{2\sigma_{y}^{2} }}} \right){ \exp }\left( { - \frac{{\left( {z - z_{0} } \right)^{2} }}{{2\sigma_{z}^{2} }}} \right),$$

(15.13)

where σ _x , σ _y , and σ _z are dispersion parameters for x, y, and z directions, respectively. The dispersion parameters increase with time. Equation 15.13 is equivalent to a solution of the diffusion equation:

$$\frac{\partial c}{\partial t} = D{\nabla }^{2} c,$$

(15.14)

where D is the diffusion coefficient of the pollutant in air. If the dispersion parameters can be converted from the diffusion coefficient as:

$$\sigma_{x} = \sigma_{y} = \sigma_{z} = \sqrt {2tD} .$$

(15.15)

Wind-blown condition

Under a wind-blown condition , the puff is considered to move along the wind velocity, u. By taking coordinates, so that the x-axis is parallel to the wind direction and assuming that the release point is (0, 0, h), for simplification, the center of the puff at a time, t, moves to (u(t), 0, h). Then, the following equation expressing the concentration of the pollutant in air is obtained by substituting the center of the moving puff into Eq. 15.13 as:

$$c\,(x, y, z, t) = \frac{q}{{(2\pi )^{{\frac{3}{2}}} \sigma_{x} \sigma_{y} \sigma_{z} }}{ \exp }\left( { - \frac{{(x - ut)^{2} }}{{2\sigma_{x}^{2} }}} \right){ \exp }\left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right){ \exp }\left( { - \frac{{\left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right).$$

(15.16)

If the wind velocity changes with time, the spatial distribution of the concentration can be obtained by substituting $(x_{0} ,y_{0} ,z_{0} ) = \int_{0}^{t} {{u}(t)\,{\text{d}}t}$ into Eq. 15.13 as well.

Reflection by the ground surface

The above-mentioned puff model, expressed as Eqs. 15.13 or 15.16, does not include the effect of interaction of the pollutant with the ground surface. In many cases of air pollutions, when the puff of the pollutant touches the ground surface, the absorption of the pollutant onto the surface is negligibly small. This means that the vertical (z) flux to the surface is zero, and therefore the concentration gradient along the vertical axis at the ground level should be zero. This condition can be reproduced by introducing an imaginary source at the inversed position of the original source to the ground surface, (0, 0, −h) as:

$$c = \frac{q}{{(2\pi )^{{\frac{3}{2}}} \sigma_{x} \sigma_{y} \sigma_{z} }}\exp \left( { - \frac{{\left( {x - ut} \right)^{2} }}{{2\sigma_{x}^{2} }}} \right)\exp \left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right)\left\{ {\exp \left( { - \frac{{\left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right) + \exp \left( { - \frac{{\left( {z + h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right)} \right\}.$$

(15.17)

The last term represents the imaginary source, and we can see that Eq. 15.17 satisfies $\partial {c}/\partial {z}|_{{{z} = 0}} = 0$. However, in the case of the water surface , HTO should be immediately absorbed without reflection different from the land surface. This must be taken into account in case of release and dispersion of T from a fusion plant as discussed in the main text.

• (b) Gaussian plume model

Assuming a steady-state release with a constant release rate from a fixed point and a stable weather condition, the pollutant can be recognized as superposition of multiinfinitesimal puffs released continuously, each of which is expressed by Eq. 15.17. Here is assumed that the concentration distribution does not change in time, different from the puff model. For simplification, the x-axis is set parallel to the wind direction and the release point to be (0, 0, h). By defining the release rate (released amount of the pollutant per unit time) as f, the released amount during an infinitesimal period from t to t + dt is fdt, and the concentration of the micro puff given during this period, dc is expressed as:

$${\text{d}}c = \frac{{f{\text{d}}t}}{{(2\pi )^{{\frac{3}{2}}} \sigma_{x} \sigma_{y} \sigma_{z} }}{ \exp }\left( { - \frac{{\left( {x - ut} \right)^{2} }}{{2\sigma_{x}^{2} }}} \right){ \exp }\left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right)\left\{ {{ \exp }\left( { - \frac{{\left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right) + { \exp }\left( { - \frac{{\left( {z + h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right)} \right\}.$$

(15.18)

We can obtain the concentration profile by integrating dc with time, as:

$$\begin{aligned} c\,(x,y,z) & = \int\limits_{t = - \infty }^{t = \infty } {{\text{d}}c} \\ & = \frac{f}{{(2\pi )^{{\frac{3}{2}}} \sigma_{x} \sigma_{y} \sigma_{z} }}{ \exp }\left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right)\left\{ {{ \exp }\left( { - \frac{{\left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right) + { \exp }\left( { - \frac{{\left( {z + h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right)} \right\}\int\limits_{ - \infty }^{\infty } {\exp \left( { - \frac{{\left( {x - ut} \right)^{2} }}{{2\sigma_{x}^{2} }}} \right){\text{d}}t} \\ & = \frac{f}{{2\pi u\sigma_{y} \sigma_{z} }}{ \exp }\left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right)\left\{ {{ \exp }\left( { - \frac{{\left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right) + { \exp }\left( { - \frac{{\left( {z + h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right)} \right\}. \\ \end{aligned}$$

(15.19)

At first glance, Eq. 15.19 does not seem to include the downwind distance (x). However, the dispersion coefficients, σ _y and σ _z , are functions of x; therefore, c is still the function of x.

• (c) Dispersion coefficient, σ

The horizontal and vertical dispersion coefficients, σ _y and σ _z , in the above-mentioned Gaussian plume model are functions of the downwind distance x and classifications of the atmospheric stability described below. The plume spreads with increasing the distance from the release origin and having stronger atmospheric turbulence. The horizontal dispersion is far greater than the vertical one. Several methods to determine the coefficients have been established. Here, two typical methods, one using reference curves and the other using approximate expressions, are introduced.

Classification of atmospheric stability

In order to determine the dispersion coefficients, it is required to evaluate how stable or how turbulent the atmosphere is. The dispersion will be small in stable atmosphere and will be accelerated in turbulent atmosphere. The stability of the atmosphere is classified by an index, “atmospheric stability class.” This index is categorized into 6 levels (A–F; F is the most stable) by evaluating wind speed and sunshine during the day or cloudiness during the night as given in Table 15.2 by Pasquill [1]. This classification is determined based on the fact that strong sunshine during daytime causes strong updraft airflow, and that clear sky during night results in stable atmosphere due to radiation cooling of ground surface.

Table 15.2 Atmospheric stability class [1]

Pasquill-Gifford curves

The first method to estimate the dispersion coefficients is referring the Pasquill-Gifford curves after determining the atmospheric stability class. The curves were graphically presented by Turner [2], and approximation formulas were proposed by Green et al. [13] as shown in Fig. 15.13.

Fig. 15.13

Pasquill-Gifford curves, a horizontal and b vertical dispersion coefficients as functions of downwind distance and atmospheric stability

Sutton’s formula

Another method, proposed by Sutton [14], is to express the dispersion parameters by power functions of the downwind distance, x, as:

$$\sigma_{y} = \frac{{C_{y} }}{\sqrt 2 }(x/{\text{m}})^{(1 - n/2)} \;{\text{m}}$$

(15.20)

$$\sigma_{z} = \frac{{C_{z} }}{\sqrt 2 }(x/{\text{m}})^{(1 - n/2)} \;{\text{m}} .$$

(15.21)

A similar method is employed in the atmospheric dispersion submodule of the NORMTRI code [10] which was developed to calculate the behavior of T released into the environment under a normal operation. In that model, σ _y and σ _z are expressed as:

$$\sigma_{y} = P_{y} \cdot (x/{\text{m}})^{{Q_{y} }} ,\quad {\text{and}}\quad \sigma_{z} = P_{z} \cdot (x/{\text{m}})^{{Q_{z} }} ,$$

(15.22)

where P _y , Q _y , P _z , and Q _z are shown in Table 15.3 with the atmospheric stability, assuming the release height of 100 m and smooth and uniform ground surface.

Table 15.3 Dispersion parameters depending on the atmospheric stability class

Maximum ground concentration

One of the most interesting points when designing the site boundary of a fusion power plant may be how high the maximum T concentration in the atmosphere at the ground level will be and where the maximum concentration will be observed. These issues can be solved by using the Gaussian plume model as follows. The maximal concentration on the ground surface must appear on the x-axis. Assuming that the dispersion coefficients are given by Eq. 15.22, and substituting Eq. 15.22 and y = z = 0 m into Eq. 15.19, the concentration at the ground level along the x-axis is presented by:

$$c\,(x, 0, 0) = \frac{f}{{\pi uP_{y} P_{z} (x/{\text{m}})^{{(Q_{y} + Q_{z} )}} }}{ \exp }\left( { - \frac{{h^{2} }}{{2P_{z}^{2} }}(x/{\text{m}})^{{ - 2Q_{z} }} } \right).$$

(15.23)

In the case of the stability class of “D,” for example, the concentration at the ground-level changes with the downwind distance for different release heights is shown in Fig. 15.14. As the release point becomes higher, the maximum of the concentration decreases, while the downwind distance x giving the maximum becomes longer.

Fig. 15.14

Concentration along x-axis for different release heights when the atmospheric stability is class D

The above-mentioned Gaussian puff and plume models are primitive; however, they have an advantage that the concentration in the atmosphere at an arbitrary point can be calculated uniquely without iterative computations. They have been used to calculate the distribution of air pollutants for long time and are still used with appropriate modifications.