Scaling laws, blast waves

Scaling of the properties of blast waves from explosive sources is a common practice, and anyone who has even a rudimentary knowledge of blast technology utilizes these laws to predict the properties of blast waves from large-scale explosions based on tests on a much smaller scale. Similarly, results of tests conducted at sea level ambient atmospheric conditions are routinely used to predict the properties of blast waves from explosions detonated under high altitude conditions. 

The most common form of blast scaling is Hopkinson-Cranz or "cube-root" scaling. This law, first formulated by B. Hopkinson (Reference 8) and independently by C. Cranz (Reference 9), states that self-similar blast waves are produced at identical scaled distances when two explosive charges of similar geometry and of the same explosive, but of different sizes, are detonated in the same atmosphere. It is customary to use as a scaled distance a dimensional parameter,... 

The blast scaling law which is almost universally used to predict characteristics of blast waves from explosions at high altitude is that of Sachs (Reference 10). Sachs law states that dimensionless overpressure and dimensionless impulse can be expressed as unique functions of a dimensionless scaled distance, where the dimensionless parameters include quantities which define the ambient atmospheric conditions prior to the explosion. 

Both scaling laws apply to reflected blast wave parameters, as well as side-on parameters. (Note that, if charge weight W is used instead of energy E, these parameters have dimensions.)... 

The Hopkinson-Cranz scaling law described earlier applies to scaling of reflected blast wave parameters just as well as it does to side-on waves. That is, all reflected blast data taken under the same atmospheric conditions for the same type of explosive source can be reduced to a common base for comparison and prediction. Sachs law for reflected waves fails close to high explosive blast sources but it does apply beyond about ten charge radii. 

Then, the calculated energy is converted in an equivalent amount of TNT, which is used to estimate shock wave effects, using the TNT equivalency method. This is based on the Hopkinson s law as scaling method to evaluate the blast pressure profile as a function of the normalized distance (AlChE/CCPS, 1994). 

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