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  • Does knowing the chemical structure tell us anything useful about an explosive?

    Does knowing the chemical structure tell us anything useful about an explosive?

    You can calculate a lot about an explosive, relatively accurately, by just knowing its chemical structure. I will describe the steps that you would need to take to get there in broad outline, but the main point to takeaway here is that it is possible. If you really want to nerd out, the process gives you a bit of insight into what kinds of explosives have what properties.

    When you first learn about different explosives they are presented to you in the form of tabulated data; “here is PETN, it has a velocity of detonation of 8,400 m/s, it is a secondary explosive. Here is RDX, it has a velocity of detonation of 8,750 m/s, its brisance is…” To start with, this is probably enough information to employ different explosives in different roles. Perhaps your job does not require you to know any more, but if you have ever asked yourself if you could figure out some of these properties by just knowing what the molecule looks like, then this is for you. The following is summarised from Explosives Engineering by Paul Cooper, Chapter 5.

    Before I describe the process, I should note that the equations can at times be long winded or have funny symbols, but they are all incredibly simple to just plug numbers into a calculator. Some steps require counting oxygen atoms and figuring out if the molecule is fuel/oxidiser balanced. Maybe you need to know a few unusual terms like “aliphatic.” But with a bit of terminology, this process is actually quite simple.

    The process goes something like this: first you calculate the theoretical maximum density (TMD) for which you need to know what kind of structure it has and the percentage weight of hydrogen. With that and a specific chemical structure you can calculate the VOD at TMD. You can vary the density to get the VOD at lower densities with the calculated TMD VOD and some empirical constants that can be set to defaults. If you have a mixture you can calculate the detonation velocity by using how much of each explosive (or inert filler, or even air) each of the components occupies. Finally, with whatever detonation velocity you have for your explosive, you can calculate the detonation pressure with a simple formula that takes density, VOD, and an empirical constant that can also be set to a default. All of this gives you single digit % accuracy for most explosives at the end. All it takes is knowing the chemical structure.

    PropertyEquationNeed to knowNotes
    Theoretical Maximum Density (TMD)\(TMD =\) \( a_{i}-k_{i}H\)– Percentage of Hydrogen by weight
    – \(a_{i}, k_{i}\) which are structural group constants which you can look up for each type of structure    		– 2-3% error
    – Hydrogen weight 0<H<6%
    Velocity of Detonation at TMD (\(D’\))\(D’ = \frac{F-0.26}{0.55}\)– Aromatic or not
    – Solid or liquid
    – Numbers of different atoms
    – Number of carbon-oxygen double bonds
    – Number of nitrate ester groups
    – Molecular weight    		– The form factor F is calculated from the variables mentioned and the equation was omitted because it looks complicated even though it is plug and play
    Velocity of Detonation at Lower Density \(D\)\(D = a + b\rho\)– Actual explosive density
    – Empirical constants \(a, b\) OR \(D’\) and \(b\), with \(b \approx 3\)
    Velocity of Detonation of Mixture\( D_{mix}=\sum_{i}D_{i}V_{i} \)– Volume ratio \(V_{i}\) of each component.
    		– Can be used even if the component is inert, filler, or even air.
    – Can be used as alternative to the above calculation of lower density VOD.
    Detonation Pressure
    (Also known as Chapman-Jouguet [CJ] Pressure)    		\(P_{CJ}= \frac{\rho D^{2}}{\gamma + 1}\)– Density and VOD of explosive
    – The ratio of specific heats of detonation product gases \(\gamma \).    		– \(\gamma \approx 3\) for typical explosives with densities \(1<\rho<1.8\).

    Let’s apply this to RDX to understand what kind of information this can give us and what kind of error sizes we get when compared to experimentally verified values:

    Image taken from Wikipedia: RDX – Wikipedia

    The first step is to find the theoretical maximum density (TMD), which since RDX is a Group 7 (solid nitrazacyclane or nitrazaoxacyclane) means \(a_{7} = 2.086, k_{7} = 0.093\), and gives:

    \( \rho_{TMD} = 1.833 g/cm^{3} \)

    This is pretty close to the literature value of \(1.816 g/cm^{3}\), or 0.94%. From this we calculate the velocity of detonation at TMD (\(D’\)) which, after skipping over the messy looking step where we count atoms and certain by-products, gives us \(F =5.177\) so:

    \(D’ = \frac{F-0.26}{0.55} = 8.94 km/s\)

    This is not immediately able to be verified experimentally because the theoretical maximum density is not really achievable, so the next step we take is to use this to calculate the VOD at the tested \(1.76 g/cm^{3}\). This gives us:

    \(D = D’ -3\Delta\rho=8.72 km/s\)

    The literature value for the VOD at that density is \(8.75 km/s\) meaning the error here is -0.32%. If were to use our calculated value to calculate the pressure of RDX:

    \(P_{CJ} = \frac{\rho D^{2}}{4} = 33.46 GPa \)

    The literature value is \(33.46 GPa\), which means our error is -1.02%. That’s pretty close.

    The short of this is that we have some pretty simple tools that can get surprisingly accurate estimates of some key performance parameters of an explosive with only the chemical structure and some generic constants. Whilst using literature tabulated values will always be easier where they exist, when dealing with novel explosives, these calculations can prove invaluable.

  • Detonations and Deflagrations, High and Low Orders

    Frequently on an explosives range you will hear people comment that a “deflagration” occurred which stopped the expected effect, or a “partial high order” meant that some of the explosives went flying instead of detonating. Do people know what they’re talking about? Usually they could stand to be a bit more precise and this post is aimed to help with exactly that.

    Detonation vs Deflagration

    Detonations and deflagrations end up with the some products but how they get there is markedly different. Basically, a deflagration is a (usually very quick) burning; you get deflagration in an internal combustion engine car to move the piston or the barrel of a gun to fire a bullet. Often deflagration is associated with propellants and pyrotechnics (previously called “low explosives”) because they are designed to undergo that quick combustion but you can cause deflagration in a high explosive too. Deflagration is fundamentally a thermal reaction in that the decomposition of the material happens at the rate at which it heats up and decomposes which happens at the surface and below the speed of sound.

    Detonation is quicker. Specifically, the rate of that decomposition is faster than the speed of sound and is associated with a shock wave.

    When do people get confused?

    With these definitions in mind, where do the lines get blurred? Most of the confusion happens when detonations or deflagrations do not go to plan and a deflagration-to-detonation or detonation-to-deflagration transition occurs. On either end of the spectrum, a full detonation is one where all the explosive is consumed by the detonation and a full deflagration is one where all the explosive is consumed by deflagration (burning). Whilst I have never heard the term used, if you fired an artillery shell and not all the propellant was consumed, you would have had a “partial deflagration”.

    What I hear more frequently is people using the term “partial detonation” and deflagration interchangeably, and I understand why the two get confused: often when a shockwave does not have enough energy to continue the chain reaction and consume all the explosive, the remaining explosive will be thrown by the blast wave, some of which will in turn burn due to the heat of the product gases. So when a partial detonation occurs, some deflagration usually occurs too. But deflagration is just burning, it is not the process of flinging around explosive materials.

    To emphasise: the event that has occurred in a partial detonation is a detonation-to-deflagration transition because some explosives were burnt up after the shock wave failed to continue the chain reaction, but the thrown around explosive material that remains at the end has neither detonated (been decomposed by the shock wave) nor deflagrated (been decomposed by burning).

    Burning to Bang: the Deflagration to Detonation Transition

    If you have followed to this point you know that deflagration is just burning, so this is the same as “burning to detonation.” To understand when this occurs we need to recall that deflagration happens subsonically at the speed of burning at the surface and detonation occurs supersonically at the speed of the shock wave. So how can we go from the rate of burning to a shock wave?

    The answer is that the rate of burning changes depending on the environment, especially the pressure. If you were to take gunpowder and light it in the ground, it would burn far slower than confined in the barrel of a gun. At some point of confinement, the rate of burning will hit a point at which it will exceed the speed of sound and a shock wave will be generated. That is how explosives burn to detonation.

  • TNT Equivalency Factors Are Not Very Good

    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.

    Figure 1: Variations of TNT Equivalency across various sources. [1]

    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:

    1. 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.
    2. 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.
    3. 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).

    Table 1: Varying TNT Equivalencies depending on the specific aspect measured. [3]

    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.

    Figure 2: TNT equivalence at different scaled distances for peak pressure. [4]

    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.