Overheating

This page summarizes the mathematical models used by ACE Overheating and shows which parameters are exposed in the server config settings.

For the numerical methods used to apply the equations below, see General Considerations.

Simulation of Barrel Temperature

The temperature of the barrel is simulated based on heat gained by fired bullets and heat lost by air convection and radiation. The barrel is approximated as a cylinder with uniform body temperature.

Heating

The heat gained by the barrel per shot $Q_{\mathrm{shot}}$ is proportional to the bullet’s initial kinetic energy $E_{\mathrm{k}\mathrm{i}\mathrm{n},0}$:

$$Q_{\mathrm{shot}}={\lambda }_{\mathrm{heat}}\cdot E_{\mathrm{k}\mathrm{i}\mathrm{n},0}$$ $$E_{\mathrm{k}\mathrm{i}\mathrm{n},0}=\frac{1}{2}m{v}_0^2$$

where $v_0$ is initial bullet speed, $m$ is the mass of the bullet and ${\lambda }_{\mathrm{heat}}$ is a scaling factor given by the product of the prefab’s m_fHeatingScale and the mission header’s m_fHeatingScale.

Passive Cooling

Heat loss is simulated with Newton’s law of cooling (for convective channels) and Stefan–Boltzmann law (for radiative channel):

$$\frac{d{Q}_{\mathrm{loss}}}{dt}={\dot{Q}}_{\mathrm{loss}}=A_b\cdot [h\cdot (T_b-T_{\mathrm{sur}})+{\epsilon }_b\cdot \sigma \cdot (T_b^4-T_{\mathrm{sur}}^4)]$$ $$A_b=\pi \cdot d_b\cdot l_b$$

where $A_b$ is the area of the lateral surface derived from barrel diameter $d_b$ and length $l_b$, $h$ is the heat transfer coefficient, $T_b$ is the barrel’s temperature, $T_{sur}$ is the temperature of the surroundings, ${\epsilon }_b$ is the emissivity (prefab’s m_fBarrelEmissivity) and $\sigma$ is the Stefan–Boltzmann constant. The heat transfer coefficient $h$ is given by:

$$h=h_0+{\lambda }_{\mathrm{cool}}\cdot \max (12.5,4.5908\cdot \frac{v^{0.6}}{d_b^{0.4}}+9.3198\cdot \frac{r}{d_b^{0.4}})$$

where $h_0$ is the base heat transfer coefficient (m_fBaseHeatTransferCoefficient), ${\lambda }_{\mathrm{cool}}$ is the cooling scale (m_fDefaultAirCoolingScale or m_fOpenBoltAirCoolingScale), $v$ is the character’s relative air speed (i.e. accounts for wind), $r\in [0,1]$ is the rain intensity and $d_b$ is the barrel’s diameter. The air speed term is derived from the cylinder crossflow correlation for $\mathrm{Re}\in [1000,20000]$ from Ref. [1]:

$$\mathrm{Nu}=0.26\cdot {\mathrm{Re}}^{0.6}\cdot {\mathrm{Pr}}^{0.37}$$

with the air parameters in the table below:

Symbol	Value	Description
${\mathrm{Pr}}_{\mathrm{air}}$	0.714672	Prandtl number
$k_{\mathrm{air}}$	0.02614 W/(m⋅K)	Thermal conductivity
${\eta }_{\mathrm{air}}$	$1.846\cdot {10}^{-5}$ kg/(m⋅s)	Dynamic viscosity
${\rho }_{\mathrm{air}}$	1.1839 kg/m³	Density

Temperature Update

The barrel’s temperature is updated with the ordinary differential equation (ODE) below:

$$\frac{d{T}_b}{dt}=\frac{{\dot{Q}}_{tot}}{C_b}$$ $${\dot{Q}}_{tot}=f_{\mathrm{RoF}}\cdot Q_{\mathrm{shot}}-{\dot{Q}}_{loss}$$ $$C_b=c_b\cdot f_b\cdot m_w$$

where ${\dot{Q}}_{tot}$ is the total rate of change of heat, $f_{\mathrm{RoF}}$ is the rate of fire, $C_b$ is the heat capacity of the barrel, $c_b$ is the specific heat capacity (prefab’s m_fBarrelSpecificHeatCapacity), $f_b$ is the barrel mass fraction (prefab’s m_fBarrelMassFraction) and $m_w$ is the mass of the weapon.

Simulation of Cook-Off

Bullets are approximated with uniform body temperature and a first-order chemical reaction for the cook-off. Only conductive equilibration with the barrel is considered for heat transfer.

Heat Exchange with Barrel

Ammunition always start at the temperature of the surroundings. The temperature equilibration between barrel and chambered bullet is done with Newton’s law of cooling:

$$\frac{d{T}_p}{dt}=\frac{\kappa }{m_p}\cdot (T_b-T_p)$$

where $T_b$ is the barrel temperature, $T_p$ is bullet temperature, $m_p$ is bullet mass and $\kappa$ is a rate constant (m_fBarrel2BulletMassScaledTemperatureRateConstant). The change in heat from this process is neglected for the barrel, since the barrel has a far larger heat capacity than the bullet.

Cook-Off Progress

Once the bullet temperature $T_p$ reaches the autoignition temerature $T_{\mathrm{ig}}$ (m_fGunpowderAutoignitionTemperature), cook-off progress $P_{\mathrm{cook}}$ increases based on an Arrhenius rate:

$$\frac{d{P}_{\mathrm{cook}}}{dt}=\{\begin{array}{cc}{k}_{\max }\cdot \exp (-\alpha \frac{T_{\mathrm{ig}}}{T_p})& |T_p\ge T_{\mathrm{ig}}\\ 0& |T_p<T_{\mathrm{ig}}\end{array}$$

where $k_{\max }$ and $\alpha$ are fitted parameters based on data from Ref. [2]. The given ODE modelled the data better than applying Arrhenius rate at any temperature. Cook-off takes place when $P_{\mathrm{cook}}$ reaches 1.

Temperature-Dependent Effects on Weapon

Jamming

The chance that a bullet jams the weapon is given below:

$$p_{\mathrm{jam}}={\lambda }_{\mathrm{jam}}\cdot f_{\mathrm{jam}}(T_b)p_{\mathrm{jam}}\in [0,1]$$

where ${\lambda }_{\mathrm{jam}}$ is a scale factor (m_fJamChanceScale), $f_{\mathrm{jam}}(T_b)$ is a temperature curve (prefab’s m_cJamChanceTemperatureCurve). The chance is evaluated whenever a bullet is fired.

Muzzle Dispersion

The multiplier applied to the base game muzzle dispersion is given below:

$$F_{\mathrm{disp}}=1+{\lambda }_{\mathrm{disp}}\cdot f_{\mathrm{disp}}(T_b)F_{\mathrm{disp}}\in [1,\infty )$$

where ${\lambda }_{\mathrm{jam}}$ is a scale factor (m_fMuzzleDispersionScale) and $f_{\mathrm{disp}}(T_b)$ is a temperature curve (prefab’s m_cMuzzleDispersionFactorTemperatureCurve). The dispersion multiplier is related to the base game dispersion cone’s half-apex angle $\theta$ and temperature-based increase $\mathrm{\Delta }\theta (T_b)$ as below:

$$F_{\mathrm{disp}}=\frac{\tan (\theta +\mathrm{\Delta }\theta (T_b))}{\tan (\theta )}\approx 1+\frac{\mathrm{\Delta }\theta (T_b)}{\theta }$$ $$⟹{\lambda }_{\mathrm{disp}}\cdot f_{\mathrm{disp}}(T_b)\approx \frac{\mathrm{\Delta }\theta (T_b)}{\theta }$$

the approximation is given by first-order Taylor expansion for small values of $\theta$. For instance, if the base game spread $\theta$ is 0.5 mrad and for a given temperature $\tau$, it should be 1.5 mrad, the increase $\mathrm{\Delta }\theta (\tau )$ is therefore 1.0 mrad and ${\lambda }_{\mathrm{disp}}\cdot f_{\mathrm{disp}}(\tau )\approx 2.0$.

References

[1] A. Žukauskas, Adv. Heat Transfer 1972, 8, 131.

[2] A. Hameed, M. Azavedo, P. Pitcher, Defence Technology 2014, 10, 86–91.