# Interior ballistic model, Powder modeling & Sources

## Interior Ballistic Model

The interior ballistic model used in the GRT is a custom development on the basis of the information given in the book
*“Waffentechnisches Taschenbuch”* by Rheinmetall^{1)}, in the books *“Ballistik - Theorie und Praxis”*
by Beat P. Kneubuehl^{2)},
as well as *“Ballistik”* by Richard Emil Kutterer^{3)} and also be influenced by IBHVG2 “lumped-parameter model”^{4)}.

### Different burning characteristics

The various propellants or powders have different burning characteristics, i.e. depending
on the manufacturer and type, the powder burns at **different** speeds during pressure
development! Such powders behave either progressive, degressive or a specific combination
of both. The traditional terms used to advertise powder behavior – e.g. “progressive”
and “offensive” do not contradict this, because even offensive powders can be very progressive.

To differentiate burn behavior for propellant selection, creates opportunities to increase projectile acceleration in a longer barrel and keep pressures low at the same time.

## Form functions & mapping

This powder burn-up characteristic is mathematically simulated in the simulation calculation by so-called form functions. This form function mathematically represents and approximates the behaviour of the powder during the burning process.

**The mathematical representation** of a powder characteristic is based on a standardized
measurement, e.g. with a calorimeter bomb or closed vessel (pressure bomb / manometric bomb), which has a
specific volume. Several measurements are then taken and recorded with a fixed test quantity
delta of a propellant powder.

The parameters of the form functions are changed so that they model the measurement curve. Subsequently, a calibration with ballistic data must be performed, because the propellant can act differently in the dynamic behavior of a gun.

**After adaptation and calibration,** the mathematical representation does not necessarily
correspond to the original measurement data, since the propellant charge behaves dynamically
differently than in the static closed vessel. Without a closed vessel measurement, it is possible
to create an reengineered model using ballistic data including pressure equipment directly on the barrel
over a calibre spectrum and bullet weights.

The closed vessel measurement is therefore one of the most important initial data for a powder model. As a rule, powder manufacturers perform this measurement for research and quality control purposes.

The value **Ba** given in the powder data is the **start** of this curve, which is composed
of two or more sections. In contrast to the other software, **GRT** has
a **three-step** representation of the burn-up behavior which accounts for multi base propellant
behaviors and partially for additives of propellants nowadays like anti-copper-fouling and
temperature-stabilizers etc.

### Determination of powder characteristics

The calculations of internal ballistics depend on these powder characteristics. To determine
the powder characteristics there is either the possibility of thermodynamic calculation
^{5)}, or experimental determination.
The experimental determination of most powder characteristics can be performed using
a *closed vessel (manometric bomb)*.

#### The Closed Vessel

##### Pressure in the closed vessel (manometric bomb)

The basic idea behind the closed vessel is that a certain amount of propellant powder
**mc** is burnt in a fixed volume **V_mb**. At *combustion* a maximum pressure
**p_mb** depending on powder and quantity is reached.

p_mb * ( V_mb - b * mc ) = mc * R * T_ex

**mc** is the powder mass, **b** is the covolume, **R** the gas constant and **T_ex**
the explosion temperature (flame temperature) of the combustion gases (usually calculated thermodynamically).

##### Powder constant, specific energy

The *specific energy* **F_se** is obtained from the product of gas constant **R**
and explosion temperature **T_ex**:

F_se = R * T_ex

This specific energy represents the pressure energy during the burning of the powder, which is available for conversion into mechanical energy.

##### Abel's equation

The English chemist *F. A. Abel* released investigations and an equation published
in 1874, proposing to use the general equation of state at high pressures. This is
the reason, why the Van-der-Waals equation are not used here. In the case of conditions
occurring in ballistics and explosives, the intrinsic volume of the gas molecules must
be also considered (covolume **b**). If the charge density **delta** is calculated from
the ratio of powder mass **mc** and combustion chamber **V_mb** **(mc/V_mb)**, the
equation known in ballistics as *Abel's equation* is obtained:

p_mb = (delta * F_se) / (1 - delta * b)

#### Measurements

Charge density in the combustion chamber of a manometric bomb must not exceed a
maximum value in order to achieve uniform combustion from measurement to measurement
and thus an evaluable pressure curve. The maximum value itself depends on the specific
explosion heat **Qex** of the powder used.
After the investigations of *Gallwitz* experience has shown that about **delta *
Qex = 545 kcal/dm³** has to be.^{6)}

##### Raw data & processing

**In practice,** however, due to ubiquitous and unavoidable scattering, measurements
must be carried out with different charge densities. Specific energy and covolume are
then determined by linear regression. The reciprocal values of the measurements performed form pairs of values. The slope of
the regression line gives the reciprocal of the specific energy **F_se**, from which the covolume **b** can be determined.

In the measurement, the pressure is recorded as a function of time, where **z(t)** is
the proportion of the converted charge. As said, the problem here is the unavoidable scattering and noise of the signal:

The recorded data is analysed & processed aftermath e.g. using filters like Savatsky-Golay or LOWESS. The results can vary dependent on the filters and parameters used. That is the reason why the values used in a GRT powder model can be slightly different from the figures published by the manufacturer, because the manufacturer may use different filters or parameters for theyr analysis.

If the law of combustion is used instead of **z(t)**, it can be resolved according to
the product of *dynamic vivacity* and *form function* **phi(z)** (see image/diagram above, Y-axis).

( Ba * phi(z) ) / p0 = (ppunkt(t) / p(t)) * ( (1- b*z(t)*delta - (1-z(t))*delta/pc)² / (delta * F_se * (1-delta/pc)) )

**b** is the covolume, **delta** the ratio of powder mass and combustion chamber
**mc/V_mb** and **pc** the density (mass density) of the powder substance.

### Temperature Influence on Propellants

**In the GRT, the change in the combustion behaviour due to the influence of powder
temperature** is represented by temperature coefficients, which separately represent
the range below and above the default temperature of 21°C (see picture on the right,
not to scale).

The user has the possibility to change the propellant temperature from the default value within a limited range in order to take account of environmental influences.

In general, the temperature drift of the propellants is represented by generally
accepted analyses and algorithms.
^{7)}
^{8)}
^{9)}
^{10)}
Some manufacturers give temperature coefficients from specific measurements for
their propellant powders, which improves the results, otherwise the internal
default coefficients are used.

**To determine the coefficients,** the vivacity measurements are repeated at a
given temperature and then calculated as follows:

##### Cold Temperature Coefficient (tcc)

tcc = (Ba(T= +21°C) - Ba(T= -20°C)) / (21+20)

##### Hot Temperature Coefficient (tch)

tch = (Ba(T= +60°C) - Ba(T= +21°C)) / (60-21)

### Energies at the shot

The chemical energy released by the conversion of the propellant charge is essentially divided into the following quantities when the shot is fired:

- The translational projectile energy
- The rotational projectile energy
- Flow energy of the powder gases
- Internal energy of the powder gases
- Heat losses at tube, bullet and sleeve
- Gas losses, friction and acoustics (vibration behavior)
- the work against the pull-out resistance
- The work of pressing the bullet into the lands
- Energy of the recoiling weapon parts
- Energy for cycling (semi-)automatic weapons

### Losses

**The friction losses on the projectile** are represented in the algorithm by models,
where frictional resistances are primarily specified by the bullet manufacturer if available.
The experienced user is given the possibility of manual adjustment.

**The flow energy of the powder gases** can be calculated by adding a proportion of the
charge (carrying factor, “Sebert-factor”) to the bullet mass to be accelerated. As is usual
in other areas of physics, an effective mass is calculated. Also other energy losses can
be considered here in the effective mass, e.g. the energy losses by heat input.

**The gas losses due to constructive conditions** such as the cylinder gap of revolvers
can amount to **up to 20%**. They can be specified by the experienced user with the help
of a wizard.

**The portion of the gun recoil**, as well as the propulsion energy of automatic
weapons is neglected.

## Data

The caliber, bullet and propellant powder data provided by the GRT are laboriously created by the GRT development team and the community and manually entered data, whereby the data of the propellant powders are based on the measurement data provided by the respective manufacturers, as well as data which have been and will be determined by GRT laboratory and the community on the basis of its own measurements.

Special thanks go to the companies (in alphabetical order):

### Important note

**Due to manufacturing fluctuations and tolerances, it is important to compare the data
provided with the real conditions and adjust them if necessary. In particular: case volume
and bullet length must always be checked and measured. No guarantee is given for the
correctness of the data provided!**

^{1)}

^{2)}

^{3)}

^{4)}

^{5)}

^{6)}

^{7)}

^{8)}

^{9)}

^{10)}