Technical Health Physics Papers - The Mathematics of Radiation Protection

Article written by Chris Robbins of Grallator Ltd

You can also download a PDF of 'RDD Radioactive Particle Transport'

This is a paper submitted by Chris Robbins of Grallator Limited. Ionactive has been discussing with Chris the prospect of future resource that might employ the modelling and simulation of a RDD (Radiological Dispersal Device), or ‘Dirty Bomb’. This resource would be aimed at training the First Responder in our 'The R&N in CBRN' training programme and would be visual with rich multi-media. However, as shown in this paper (which Chris describes as ‘back-of-the-envelope’!), the mathematics are extensive -  but they and the physics which underpin them are essential if the model and simulation are going to be viable.

There are many models out there that are related to this area of simulation (just try google). However, if we were to use this for training the onus would be on using the maths and physics to provide realism  - but the output would be 'a first person game perspective'.

Only the first bite sized portion of this paper is featured below, please use the link above to download the full paper as a PDF.

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One of the possible scenarios for a terrorist attack is the detonation of an explosive device that is encased in radioactive material; the so-called “dirty bomb”. In this scenario fine particles of radioactive material are released into the environment which are transported by three mechanisms

• ballistics, particles are ejected in random directions and fall to earth under gravity.
• advection, i.e. being blown away by the wind.
• diffusion where very fine particles are dispersed as they travel.

The first mechanism is an important component in the determination of the so-called deposition velocity. This is the speed at which particles settle back to earth. The second and third mechanisms are the dominant method of transport for small particles. In the near-field and especially in urban environments, where buildings modify wind in a complex manner, advective transport can be complex. Over larger distance scales dispersion is often treated using Gaussian plume type approaches which include a simple wind average advective term in their formulation.

This note will develop some simple mathematical models based on the ballistics of the release to provide a background for some of the important aspects of radioactive particle dispersal. It is not intended to be a full study.

Ballistic behaviour

The ballistic behaviour of the particles can be modelled by assuming that at the scales of the particles Stokes flow is valid and that the friction on a spherical particle of radius r is given by

where m is the coefficient of viscosity for air and v is the velocity of the sphere. If the velocity of the particle is resolved into horizontal and vertical components the following equations of motion can be constructed from Newton’s Second Law.



where x and y are the horizontal and vertical displacements from an origin, m is the particle mass and g is the acceleration due to gravity. The coordinates are such that right and up are in the positive direction.

For a spherical particle of density r, the mass is given by



Substituting this into (2) yields



where



The parameter a gives the relative importance of viscous drag effects to inertial effects.

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You can read the rest of this article as a PDF here.

 

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