Analytical solutions for exposures and toxic loads in well-mixed shelters in support of shelter-in-place assessments

Understanding the exposure and toxic load for the interior of buildings during and following the passage of an external airborne hazard can be a critical piece of information in deciding the benefit from adopting a shelter-in-place strategy. Whilst numerical methods allow the calculation of such parameters for the general case, analytical solutions allow more rapid assessments to be made and highlight the key parameters more clearly. Analytical expressions are derived for the exposure due to the acute inhalation of toxic chemicals and the associated toxic load as a function of time, external hazard duration and building air change rate assuming a top-hat outdoor concentration profile and no indoor loss mechanism. It is shown that the internal exposure tends to the external exposure at long times for any external concentration profile. Expressions are derived for toxic loads with exponents n = m/2 where 2 ≤ m ≤ 7 is an integer to cover the range of typical values (1 ≤ n ≤ 3.5). At long times the ratio of internal to external toxic load for a top-hat outdoor concentration profile is shown to be a function of the product of the air change rate and the duration of the external hazard.