Radon build-up in a prototype dwelling using uranium mill tailings as construction material

Abstract

Even though the use of uranium mill tailings as a construction material is prohibitive such prototype dwellings can be constructed for developing models for the human exposure assessment. Among the potential challenges of uranium mill tailings build-up of radon (²²²Rn) from the tailings pile and subsequent migration following dispersion is assumed significance due to the associated inhalation hazard of its progeny. Sources of such exposure are predominantly considered as the ²²⁶Ra (parent of ²²²Rn) present in the construction material and underlying geophysical characteristics. In the present investigation, a model is developed presuming the use of mill tailings as the construction material of a dwelling (prototype). The numerical model prediction for 1.14 Air Changes per Hour (ACH) reflects that the level can be approximated at 300 Bq m⁻³, a reference level proposed in ICRP recommendation (ICRP 2014).

Appendix

Consider a control volume of cross-sectional area A as shown in Fig. 

Fig. 4

Control volume for FVM

4 and midpoint of each control volume is E, P, W with diffusion coefficient D_E, D_P, D_W respectively. From conservation of activity, one can write,

$$\left( {f_{e} - f_{w} } \right)A = \smallint \lambda \left( {S_{p} - C} \right){\text{dv}}$$

(9)

In Eq. (9) $f_{e} = - D_{e} \frac{{C_{E} - C_{P} }}{{{\text{dX}}_{{{\text{EP}}}} }}$ & $f_{w} = - D_{w} \frac{{C_{P} - C_{w} }}{{{\text{dX}}_{{{\text{PW}}}} }}$ are flux at e & f boundary respectively and all c’s represent radon concentration in respective control volume. The diffusion coefficients at the boundaries modified into $D_{e} = \frac{{D_{E} {\text{dX}}_{{{\text{eE}}}} + D_{P} {\text{dX}}_{{{\text{Pe}}}} }}{{{\text{dX}}_{{{\text{PE}}}} }}$ & $D_{w} = \frac{{D_{W} {\text{dX}}_{{{\text{wW}}}} + D_{P} {\text{dX}}_{{{\text{wP}}}} }}{{{\text{dX}}_{{{\text{PW}}}} }}$ respectively. The concentration at each node will follow linear coupled equation given by Eq. (10).

$$a_{P} C_{P} = a_{E} C_{E} + a_{W} C_{W} + \lambda S_{p}$$

(10)

In Eq. (10), all the coefficients are defined as, $a_{E} = \frac{{D_{e} }}{{{\text{dX}}_{{{\text{EP}}}} {\text{dX}}_{{{\text{ew}}}} }}$ & $a_{W} = \frac{{D_{w} }}{{{\text{dX}}_{{{\text{WP}}}} {\text{dX}}_{{{\text{ew}}}} }}$ $a_{P} = a_{E} + a_{W}$. These coefficients will be similar for all the internal points. For end points of the control volume, certain correction is required due to closeness of endpoints to the centre of control volume as shown in Fig. 

Fig. 5

Control volume with edges

5. For the node at atmosphere and wall boundary, the coupled equation is,

$$a_{P} C_{P} = a_{E} C_{E} + \lambda S_{p} - \frac{{2D_{w} C_{out} }}{{{\text{dX}}_{{{\text{WP}}}} {\text{dX}}_{{{\text{ew}}}} }}$$

(11)

With coefficients, $a_{E} = \frac{{D_{e} }}{{{\text{dX}}_{{{\text{EP}}}} {\text{dX}}_{{{\text{ew}}}} }}$ & $a_{W} = 0$ $a_{P} = a_{E} + \frac{{2D_{w} }}{{{\text{dX}}_{{{\text{WP}}}} {\text{dX}}_{{{\text{ew}}}} }}$. Similarly, at the boundary between wall and inside the dwelling node, the coupled equation is

$$a_{P} C_{P} = a_{W} C_{W} + \lambda S_{p} - \frac{{2D_{e} C_{2} }}{{{\text{dX}}_{{{\text{EP}}}} {\text{dX}}_{{{\text{ew}}}} }}$$

(12)

With coefficients, $a_{W} = \frac{{D_{w} }}{{{\text{dX}}_{{{\text{WP}}}} {\text{dX}}_{{{\text{ew}}}} }}$ & $a_{E} = 0$ $a_{P} = a_{W} + \frac{{2D_{e} }}{{{\text{dX}}_{{{\text{EP}}}} {\text{dX}}_{{{\text{ew}}}} }}$.

For zero flux boundary, either $f_{e}$ or $f_{w}$ will be zero and all the coefficients of linear equation will be modified accordingly. The coupled linear equation will form a tri-diagonal matrix and has been solved using TDMA.

The method of determining steady state radon concentration inside dwelling is following.

• (a) Assume the initial concentration inside c₂ room.

• (b) Use TDMA to determine the concentration at each node of numerical domain and hence estimate flux at inside wall boundary for outside concentration ${\mathrm{c}}_{0}$ and inside concentration c₂.

• (c) Update the inside concentration ${\mathrm{c}}_{2}$ using Eq. (6).

• (d) Repeat (b) and (c) till equilibrium reaches.