# The Study of Gun Barrel’s Two-Dimensional Nonlinear Thermal Conduction

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## Abstract

In order to improve the life of gun barrel influenced by periodic transient thermal shock during firing, it is necessary to establish the heat conduction model of gun barrel to study the temperature field and its variation rule. Therefore, a mathematical model of two-dimensional nonlinear heat conduction is established. The governing equations and boundary conditions are linearized by Kirchhoff’s variation, and the finite difference equations of internal nodes and boundary nodes are derived using energy balance method and alternating difference implicit scheme. Based on the numerical results of the classics interior ballistic, temperature distribution of some 12.7 mm machine gun barrel during 120 successive firing rounds under the firing specification of the GJB3484-98 is calculated numerically. The temperature field of the external surface of the barrel is tested and the variation law of the temperature field is obtained. Comparison with experimental results shows good agreement with the simulation. The research results provided scientific basic for the studies of new barrel materials and coatings.

## Keywords

ADI Gun barrel Modeling Nonlinear heat conduction Numerical simulation Temperature field## List of Symbols

- \( T \)
Temperature of gun barrel

- \( \lambda \)
Thermal conductivity of the barrel material

- \( \lambda_{0} \)
Thermal conductivity of the barrel at the temperature of 0 °C

*β*Coefficient

*t*Time

*r*Distance between the node in the barrel and the barrel axis line

- \( \rho \)
Density of barrel material

- \( c \)
Specific heat of barrel material

- \( T_{\text{a}} \)
Ambient temperature

- \( f(r) \)
Barrel’s temperature distribution along radial direction caused by fired projectiles

- \( r_{0} \)
Internal radius of barrel

- \( r_{N} \)
External radius of barrel

- \( T_{\text{g}} \)
Temperature of propellant gas in barrel

- \( h_{\text{g}} \)
Composite heat transfer coefficient between propellant gas and gun barrel

- \( h_{\text{a}} \)
Composite heat transfer coefficient between ambient temperature and gun barrel

- \( h_{\text{e}} \)
Radiation heat transfer coefficient

- \( v_{\text{g}} \)
Velocity of propellant gas

- \( \rho_{\text{g}} \)
Density of propellant gas

- \( d \)
Caliber of gun barrel

- \( \lambda_{\text{g}} \)
Thermal conductivity of propellant gas

- \( C_{\text{Pg}} \)
Specific heat capacity at constant pressure of propellant gas

- \( \mu_{\text{g}} \)
Dynamic viscosity of propellant gas

- \( \varepsilon_{\text{g}} \)
Radiation rate of propellant gas

- \( \varepsilon_{\text{F}}^{\prime } \)
Effective radiation rate of gun barrel

- \( T_{\text{r}} \)
Temperature of barrel’s internal bore

- \( T_{0} \)
Temperature of gun barrel’s internal bore

- \( T_{\text{R}} \)
Temperature of barrel’s external surface

- \( \lambda_{\text{a}} \)
Thermal conductivity of air

- \( \nu_{\text{a}} \)
Kinematic viscosity of air

- \( \alpha_{\text{V}} \)
Volume expansion coefficient

- \( C_{\text{P}} \)
Specific heat capacity at constant pressure of air

- \( \mu_{\text{a}} \)
Dynamic viscosity of air

- \( C_{1} ,\;n_{1} \)
Corresponding coefficients with Grashof number

*D*External diameter of gun barrel

- \( \varepsilon_{\text{a}} \)
Radiation rate of air

*U*No physical meaning, used to linearize the governing equations, corresponding to T

- \( \alpha \)
Thermal diffusivity of barrel material

- \( \Delta {\text{t}} \)
Time step

- \( \Delta {\text{r}} \)
Radial step

- \( \Delta {\text{r}} \)
Axial step

*i*x-directional unit node

*j*r-directional unit node

*n*t-directional unit node

## Notes

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