The TNT Equivalency Factor is an attempt to summarise the energy output and blast effects of an explosive relative to TNT. An explosive is taken and tested for how much would be needed to achieve the same effect as a kilogram of TNT and given the TNT equivalence factor. In military contexts, its most important use is for estimating safety distances for blast and fragmentation effects. Unfortunately, TNT equivalency factors have a number of flaws that make it poor at doing just that. In this article we will discuss these flaws and present a pathway for better calculations of safety distances.
The Tests and Literature Values are Inconsistent
If the aim is to summarise all the different features of an explosive in one number to compare them, it would be helpful if we could all agree on what that number is. Figure 1 shows just how much variation the TNT Equivalence can have in different sources, as much as 50% in some cases with typical variation of 20-30%. I would have failed my safety calculation exam immediately if I had regularly been out by 20-30%, which shows that something is flawed with the approach we take.
One of the biggest issues with calculating TNT equivalence that leads to this spread of values is that there is not one single test, and each of the different tests capture a slightly different aspect of an explosive’s effect as it relates to TNT. A series of the most common tests and their shortfalls are discussed by Cooper. [2] The problems identify are typically either:
- The test fails to account for the loss of energy from the explosive to the apparatus. Example: the air blast test fails to account for the energy of a higher brisance explosive in shattering the steel casing into smaller and faster fragments.
- The test fails to account for the way in which gas products are generated. Example: the ballistic mortar test does not account for non-ideal explosive effects or any explosive that continues to create mechanical energy post-detonation.
- The test is a proxy for peak pressure which can be calculated more simply without doing the test. Example: the plate dent test tracks peak pressure well enough that it is used to experimentally verify theoretical value.
Some reference tables separate two important features of explosives, namely the peak pressure and impulse, to provide two TNT equivalency factors, as can be seen in Table 1 from the US Department of Defense Explosives Safety Board. It is worth noting that, whilst some explosives are relatively consistent, others have significant differences between the pressure or impulse based equivalency factor. HBX-3 for instance is more powerful with regards to pressure but less powerful with regards to impulse than TNT. The number cited more regularly is the peak pressure number (left hand column in Table 1).
These inconsistencies between different tests and sources are challenging but in principle solvable if we all agreed on a specific testing protocol. However, the problem runs significantly deeper.
TNT Equivalency Factors Do Not Scale
Once the TNT equivalency of some explosive is calculated we should be able to continue our calculations as if we had that amount of TNT instead of the original explosive. Unfortunately the next hiccup in this process is a lot harder to solve: the TNT equivalence factor is different at different distances. This is ultimately because the blast wave characteristics of different explosives are different. Figure 2 shows how the equivalency factors scale at different distances for a few different explosives.
What is important to note from this graph for our purposes is twofold:
1) The factor that would need to be used at different Z (also known as K) factors is different. So to calculate a safety distance for a K factor of 20 you would need to use a different TNT equivalence factor than for a calculation at a K factor of 100.
2) The shape of the curves is different for different explosives. That means that the scaling effect cannot be accounted for by just adding a scaling constant, it would change depending on which explosive is being used. Bespoke calculators with these curves pre-programmed would need to be used or extensive tables to look up values for each explosive at each distance.
In short, because TNT equivalency factors vary with different scaled distances and have different variations for different explosives, a constant factor cannot be used to simplify a calculation without introducing significant error.
Even Worse for Fragmentation Effects
The challenge with using TNT equivalence is significantly worse for fragmentation as more factors come into play in terms of the creation and trajectory of fragmentation. TNT equivalence factors are only derived for blast.
We noted earlier that some of the tests for TNT equivalency, specially the reliable ones, are proxies for peak pressure. When it comes to fragmentation that becomes a problem for deriving accurate safety distances for fragmentation because higher peak pressures change the characteristics of fragmentation significantly in a few ways.
Energy is transferred to the casing of a munition by three modes: shock heating, strain and fracture, and kinetic energy of the fragments. [2] Each of these steps will take different amounts of energy depending on the peak pressure of the blast wave: higher peak pressure produces greater shock heating, more strain to the casing resulting in smaller fragmentation from the fracture, and higher fragmentation velocity. In turn the size of the fragmentation impacts the distance they travel. Since these features are poorly capture in a TNT equivalence factor, the fragmentation effects are also poorly estimated from calculations based on TNT equivalence factors.
Improving Blast and Fragmentation Calculations
First of all, it is clear that blast and fragmentation effects need to be modeled with different factors to understand resulting safety distances. For blast, Locking proposed using the Power Index (PI) as a factor to account for both the heat produced and work available. [4] The PI is based on the explosive power, which is the amount of energy available that can produce an effect on the surroundings. The explosive power is converted to a PI by taking the ratio to the explosive power of TNT. Locking’s paper concludes that Power Index is the most reliable factor for modelling blast effects which in turn means it will be the most useful factor for easily generating blast safety distances.
As for fragmentation, the two factors which dictate the spread are the velocity of the fragments and the effects of drag. These will vary tremendously not only based on the explosive but also the fragmentation; from the ball bearings in resin of a grenade, to the fragmentation of a steel cased bomb, to the sections of a casing that do not fragment and as projected whole. In principle, all these factors need to be taking into consideration for accurate fragmentation safety distances.
Some balance between ease of calculation and accuracy of the end result needs to be found to efficiently keep personnel and property safe without overly burdensome safety margins. A good compromise between ease of calculation, safety, and avoiding excessive safety margins is fragmentation safety distances based on the Gurney equations.
The Gurney equations derive from work by Ronald Gurney in the 1940s and are accurate enough for calculating safety distances without needing more advanced computer models. They are based around different kinds of explosive geometry – cylindrical, spherical, sandwich, etc. – and for most military applications are more than enough to provide safe and accurate safety distances. The literature values needed for a calculation based on Gurney equations are the velocity of detonation and Gurney energy of an explosive, both of which can be easily tabulated for quick reference. In fact, whilst it is outside the scope of what can be reasonably done during a high pressure task, Gurney energy can be estimated from theory even for unknown explosives.[5]
Regardless of the calculation used, it is important to keep in mind the limitations of using traditional TNT equivalency factors to accurately model either blast or fragmentation effects of explosives.
References
[1] R. Cheesman, “Definition and Use of TNT Equivalencies in Assessing Loading in Confined Structures,” in ICPS Conference, Manchester, 2010.
[2] P. Cooper, “Comments on TNT Equivalence,” in International Pyrotechnics Seminar, Colorado Springs, 1994.
[3] Department of Defense Explosives Safety Board, Blast Effects Computer- Open User’s Manual and Documentation, Alexandria, Virginia: DDESB, 2018.
[4] P. Locking, “The Trouble with TNT Equivalence,” in 26th International Symposium on Ballistics, Miami FL, 2011.
[5] D. Frem, “A mathematical model for estimating the Gurney velocity of chemical high explosives,” FirePhysChem, pp. 281-291, 2023.
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