Thermodynamic Efficiency of the .300 Hawk Cartridge

 by Alvin Byars, AEM Enterprises, Inc.

Fred Zeglin, the gun guru at Z-Hat Custom has often made the comment that his Hawk line of cartridges were among “the most efficient cartridges available.”  Not ones to take boasting lightly, we at AEM Enterprises, Inc. decided to look closely at his claim.  It turns out that Fred is right.  And here’s the proof.

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The Laws of Thermodynamics

Before the math and science in this topic turn you off, please read on.  Our intent is to lightly cover the immutable laws that govern how efficient any energy conversion can be. After all, that’s what we’re examining.

Classical Thermodynamics is the science that studies energy and the transformation of energy into work, or moving things around.  When we speak of the thermodynamic efficiency of a cartridge, what we mean is the efficiency of the cartridge in transforming the chemical energy contained in the unfired propellant into the kinetic energy of the speeding bullet.  Classical Thermodynamics is completely described by two laws.

The first law is often referred to as the conservation of mass and energy.  Mass (or matter, the stuff of which things are made) and energy can be transformed from one into the other.  But, the total amount of mass and energy after any transformation is the same as the total we started with before the transformation.  Energy and matter cannot be created or destroyed.  They can only be changed from one form to another.  This law also tells us that when we completely burn one pound of gunpowder, we are going to end up with one pound of gas.  When we ignite the powder in a cartridge and release its energy, only part of the energy will be transformed into the bullet’s motion.  What doesn’t get transformed into bullet motion is going somewhere, either barrel heat and wear, or hot gases, or unburned powder.  But, none of it is just going to disappear.  Another way of stating the first law of thermodynamics is “there’s no such thing as a free lunch.”  This is common sense stuff.

The second law is even more common sense.  It says that energy will always flow from high-energy sources to low energy sources.  We are never going to get warm by hugging something colder than we are.  Gunpowder is high-energy stuff.  By comparison, a bullet moving down a barrel is low energy.  When we ignite the gunpowder, the bullet is going to move.  It’s not going the other way.  The second law also sets a limit on the transformation of energy.  When we set something hot right up next to something cold, heat is going to flow from the hot thing to the cold thing until both things are the same temperature.  Then the heat stops flowing between the two things.

The “model” most often used by thermodynamics to describe the transformation of heat energy into energy of motion is a piston in a cylinder being pushed by hot expanding gases.  In this model, heat is injected into the cylinder, the pressure of the gas increases, the rising pressure moves the piston and increases the volume of the cylinder.  As the volume of the cylinder increases, the pressure decreases and the temperature drops.  Thermodynamics represents this model mathematically with the following equation:

            P * V = n * R * T       where   P is the pressure of the gas,

                                                            V is the volume of the gas,

                                                            n is the quantity of gas,

                                                            R is a number called the Universal Gas Constant,

                                                            (about 1.25 for smokeless propellants), and

                                                            T is the temperature of the gas.

Thermodynamics calls this the Equation of State.  This model described by the Equation of State is completely analogous to the pressure of hot gases from ignited gunpowder pushing a bullet down the barrel of a gun.

You can relax now, because the above equation is the last one you are going to see in this article.  Everything presented in the rest of this article can be derived from this equation.  The important things to remember are these:  The first law tells us that when we burn a pound of powder, we are going to get a pound of gas.  The Equation of State tells us that as the temperature rises and more gas is produced, the pressure rises and pushes the bullet down the barrel.  As the bullet moves down the barrel, the volume of gas behind the bullet increases, and the pressure and temperature drop.  Finally, the second law tells us that chemical energy in the gun powder will continue to be transformed into increased bullet velocity until the gas pressure, volume and temperature reach some kind of equilibrium.

Thermodynamics and Cartridge Efficiency

One of the questions asked by thermodynamics is, “How much energy can I convert into work?”  In our case, “What is the maximum velocity I can get from this cartridge?”  Without going into a lot of mathematics, suffice it to say that it depends on only two things:  the energy contained in the gunpowder, and the proportion of that energy that is converted into bullet motion.  Common sense, isn’t it?

The amount of chemical energy contained in a pound of pure gunpowder is pretty much the same from one gunpowder to the next, around 1,550,000 foot-pounds.  The thing that changes from one gunpowder to the next is the burning rate.  The shape of the powder and the coatings, deterrents and other additives determine the burning rate.  These additives make up about 15% of the powder as it comes from the canister, so that the chemical energy contained in a pound of gunpowder out of the canister is about 1,325,000 foot-pounds.  That is roughly 189 foot-pounds per grain of powder.  For example, a cartridge that holds 75 grains of powder holds about 14, 175 foot-pounds of chemical energy.  If you take a look at Figure 1, you will see the chemical energy contained in five popular .308 caliber cartridges.  This chart assumes that each cartridge has the bullet seated one caliber deep, that the case is completely filled with powder, and that the density of the powder is 0.85, which is typical for rifle powders.  It is important to understand that when we speak of density here we are talking about the density of the powder itself, not the load density.  In this case, the load density is 100%.  The powder density (more correctly, the specific gravity or specific weight of the powder) is assumed to be 85%.

Figure 1

The chart in Figure 1 is really nothing more than the case capacity of the five cartridges.  However, rather than showing case capacity in terms of how many grains of water each case can hold, it is shown in terms of how many foot-pounds of chemical energy each case can hold when filled with gunpowder.  Notice in the chart that the .300 Hawk is pretty average in its “energy capacity” as far as .308 caliber cartridges go, not much greater than the .30-06 Springfield.

Now, if all of these .308 caliber cartridges were equally efficient in converting chemical energy into projectile motion, then we would expect the .300 Hawk to be fairly average in terms of bullet velocity since it is fairly average in terms of energy capacity.  However, take a look at Figure 2.

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 Figure 2

Figure 2 shows the average velocity for the five .308 caliber cartridges with various bullet weights.  These velocities are the average maximum velocities from several reloading manuals rounded off to the nearest 25 feet per second.  As you can see, the velocities produced by the .300 Hawk are very close to those produced by the .300 Winchester Magnum, and right on the heels of the .300 Weatherby Magnum.  Obviously, not all cartridges are equally efficient in converting chemical energy into bullet velocity.

Perhaps even more interesting than the average velocities for the five .308 caliber cartridges are the kinetic energies.  After all, that is what we are really interested in when we talk about cartridge efficiency.  How efficient is the cartridge in converting the chemical energy in the gunpowder into kinetic energy of a speeding bullet?

Figure 3

Two interesting things can be seen in Figure 3.  First, the average amount of kinetic energy produced by a given cartridge loaded to its maximum is pretty much the same regardless of bullet weight.  Second, the kinetic energy produced by the .300 Hawk is very close to that produced by the .300 Winchester Magnum, and again right on the heels of the .300 Weatherby Magnum.

So, I can hear you asking, “What determines how efficient a cartridge is in converting chemical energy into bullet velocity?”  Thermodynamics tells us that it is determined by three things:

1)       Volumetric expansion ratio,

2)      Heat losses, and

3)      Pressure gradient in the barrel.

The volumetric expansion ratio is nothing more that the total volume of the cartridge case plus the volume of the barrel, divided by the volume of the case.  Figure 4 shows the volumetric expansion ratios for our five .308 cartridges.

Figure 4

The expansion ratios shown in Figure 4 are based on 24 inch barrels.  The graph is sort of the opposite of the case capacities.  In general, everything else being equal, smaller cartridges are more efficient and longer barrels are more efficient.  This is all based on the Equation of State.  But, the Equation of State describes an “ideal closed system” from which no energy can escape.  A cartridge igniting in a gun is a pretty open system and a heck of a lot of energy is lost to heat.  Below is shown the conversion of chemical energy that takes place in a typical rifle cartridge when the gunpowder is ignited:

          Mechanical energy

Projectile motion           32%

Barrel friction                2%

Thermal energy

Hot gases                     34%

Barrel heat                   30%

Chemical energy

Unburned propellant       1%

As we can see, nearly two thirds of the chemical energy in the gunpowder goes to heat, either a hot barrel or hot gases.  This is clearly illustrated in Figure 5.

Figure 5

The interesting thing to note in Figure 5 is that as we continue to increase cartridge size and chemical energy capacity in an attempt to get more velocity and kinetic energy, the decreasing expansion ratios quickly catch up with us and we reach a “point of diminishing returns.”  Beyond that point we produce a lot more heat, but not much more velocity and kinetic energy.  Hence, heat loss is the second thing that determines cartridge efficiency.

The third thing that determines cartridge efficiency is the pressure gradient in the barrel.  Figure 6 shows the estimated pressure gradient for the .300 Hawk with a 180-grain bullet leaving the muzzle at 3,000 feet per second, as graphically produced by the AccuLoad precision reloading program. 

Figure 6

As can be seen in Figure 6, pressure in the barrel quickly rises, reaching a maximum when the bullet has traveled a bit less than two inches, then gradually decreases until the bullet exits the barrel.  The longer we can keep the pressure up in the barrel, the more chemical energy we will convert to kinetic energy and bullet velocity.

Now, let’s see if we can wrap all this up by taking a different view of the information shown in Figure 5.  When we talk about the thermodynamic efficiency of a cartridge, we are really talking about the percentage of chemical energy contained in the cartridge’s gunpowder that will be converted into kinetic energy of a speeding bullet.  This is exactly what is shown in Figure 7. 

Figure 7

One thing is very clear from Figure 7.  The .300 Hawk cartridge is considerably more efficient than other smaller or larger .308 caliber cartridges, based on the effective use of energy conversion.  

Defining efficiency varies from one source to another; Pejsa defined it as a ratio of energy delivered at the muzzle as compared to the energy wasted in recoil.  No mention was made in the examination of energy loss due to heat, etc. and the honors are always pointed toward the heaviest firearm.  Another popular theory for efficiency is the ratio of velocity gained per grain of powder used.  The major problem with this method is that the scale is tipped toward the smallest cartridge and bullet. In other words, the old .22 Rimfire Short will usually come out on top.  As we have shown, efficiency can be based on the most effective use of the energy contained in the gunpowder.  And in this light, the .300 Hawk really shines.

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