**2. Initial Boundary Value Problem**

The results of the transient heat transfer numerical simulations in the wall of a 35 mm cannon barrel for a single shot and for a sequence of shots has been presented in this paper. The initial temperature of the cannon was assumed as *T*<sup>0</sup> = 20 ◦C. The heat transfer on the barrel's outer surface was modeled as a boundary condition of the 3rd kind in a form . *q* = *hout*·(*T*(*t*,*rz*, *z*) − *T*0). An equivalent heat transfer coefficient value of *hout* = 9.2 W/ m2·<sup>K</sup> was assumed to be the same on the entire outer surface of the barrel. The governing equation for nonlinear and axially symmetrical 2D IBVP is as follows:

$$
\rho\_s(T)c\_s(T)\frac{\partial T}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r}\left(k\_s(T)r\frac{\partial T}{\partial r}\right) + \frac{\partial}{\partial z}\left(k\_s(T)\frac{\partial T}{\partial z}\right) \tag{26}
$$

with

$$r\_{in} < r < r\_{out}, 0 < z < l\_{m\prime} \text{ t } > 0,\tag{27}$$

where *T* is the temperature of the gun barrel, *t* is the time, *r* is the distance between node and the barrel axis line, *ρ* is the density of barrel material, and *c* is the specific heat of barrel material, with the initial condition:

$$T(0, r, z) = T\_0 \text{ with } r\_{\text{in}} < r < r\_{\text{out}} \, 0 < z < l\_{\text{in}} \text{ and } t = 0 \tag{28}$$

and the boundary conditions:

$$\dot{q}\_i(t, r = r\_{\rm in}, z) = h\_i(t) \cdot (T(t, r\_{\rm in}, z) - T\_{\rm g}(t, r\_{\rm in}, z)), \; i = 1, \; \dots \; , 6,\tag{29}$$

(*i*–a zone number from S1 to S6)

$$\dot{q} = h\_{out} \cdot (T(t, r\_{out}, z) - T\_0)\_\prime \tag{30}$$

where *Tg* is the gas temperature calculated by solving the internal ballistic model, *rin* = <sup>35</sup> 2 mm, *rout* dependent on the variable *z*–Figure 1.

The same IBVP was solved for the series of shots (27) to (30). The initial condition for the next shot was taken from the previous solution, i.e., *T tj*,*r*, *z* = *T*(0,*r*, *z*), with *j* standing for the number of shot. The boundary conditions remained unchanged during calculations. The calculations were made using FEM implemented in the COMSOL Multiphysics program. The number of mesh elements, including quad elements, is 26,200. Minimum element quality equals 0.8563. Duration of a single shot was 100 ms. A sequence of shots was adopted for the simulation of burst firing. The calculations were made on 6 sections of the barrel S1 to S6 (*z* in the middle of each zone). The numerical experiment was carried out on the DELL PRECISION TOWER 5610 workstation equipped with an Intel (R) Xeon (R) CPU ES-1620 v3 @ 3.50GHZ with 16 GB RAM under Windows 10 Operating System. The total computation time of the sixty shots was approximately 6 h. Mesh compaction near the inner surface was performed with a geometric sequence with length element ratio equaling 0.81 and the mesh between the zones compacted five times—Figure 6.

**Figure 6.** Meshed cell with quad elements of a barrel (shown in Figure 1).

#### *2.1. Temperature Distibution in the Cannon Barrel for a Single Shot*

For each of the selected steels, the temperature distributions *Ti*(*t*,*rin*, *z*) of the barrel's inner surface at the 6 zones S1 to S6 (*z* in the middle of each zone) for the single shot are shown separately in Figure 7. In each zone, the heat transfer coefficient as a function of time *hi*(*t*) is different—Figure 2. The dashed line on each Figure shows the time the bullet left the barrel (*t* = 4.54 ms).

**Figure 7.** Temperature distribution *Ti*(*t*,*rin*, *z*) of the barrel's inner surface at the 6 zones S1 to S6 (*z* in the middle of each zone) for the single shot for the selected steels. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

The highest temperature, i.e., the so-called highest peak temperature occurs for DU-PLEX steel. The 38HMJ and 30HN2 MFA steels behaved similarly, i.e., the temperature distribution *Ti*(*t*,*rin*, *z*) of the inner surface of the barrel in the six zones S1 to S6 (*z* in the middle of each zone) were practically the same for one shot. For each selected steel, zone S3 had the highest temperature (Figure 8). The disturbance of the temperature distribution *Ti*(*t*,*rin*, *z*) of the inner surface barrel in zone S5 was caused by a rapid decrease in the heat transfer coefficient *hi*(*t*) at the moment the bullet left the barrel—Figure 1. In zone S6, this effect did not occur because the bullet travelled there too briefly.

**Figure 8.** Temperature distribution, the so-called highest temperature *Ti*(*t*,*rin*, *z*) of the barrel's inner surface in the 6 zones S1 to S6 (*z* in the middle of each zone) for the single shot for each selected steel, separately. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

#### *2.2. Temperature Distibution in the Cannon Barrel for a Series of Seven Shots*

In all the presented calculations, we assume the time-dependent heat flux density on the inner surface of the barrel changes for the first and subsequent shots. This is because the temperature of the inner surface of the barrel changes. For the selected steels, the temperature distribution *Ti*(*t*,*rin*, *z*) of the barrel's inner surface at the six sections S1 to S6 (*z* in the middle of each zone) for the series of seven shots is shown in Figure 9.

**Figure 9.** Temperature distribution *Ti*(*t*,*rin*, *z*) of the barrel's inner surface at the 6 zones S1 to S6 (*z* in the middle of each zone) for the sequence of seven shots for the selected steels. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

The lowest temperature, i.e., the lowest peak temperature, for 38HMJ and 30HN2MFA steel was the same for each shot in a series of seven shots. However, in the case of DUPLEX steel, this temperature was higher for each shot in a series of seven shots compared to the 38HMJ and 30HN2MFA steel.

#### *2.3. Temperature Distibution along the Barrel Thickness for a Series of Seven Shots*

For the selected steels, temperature distributions *Ti*(*t*,*r*, *z*) along the barrel thickness for a sequence of seven shots in zone S6 (*z* in the middle of the sixth zone) and for the for the first, fourth and seventh shots in zone S6 are shown in Figure 10. In addition, the temperature distributions *Ti*(*t*,*r*, *z*) along the barrel thickness for the selected steels for the first, fourth and seventh shots separately are illustrated in Figure 11.

**Figure 10.** Temperature distribution *Ti*(*t*,*r*, *z*) along the barrel thickness for selected steels for a sequence of seven shots: left side—*z* in the middle of the sixth zone S6 (color-coded for the distance from the inner surface of the barrel); right side—for the first, fourth and seventh shots. The sign: P6.5—in the middle of the S6 zone.

Figure 11 shows the temperature distributions *Ti*(*t*,*r*, *z*) along the barrel thickness for a sequence of seven shots in zone S6 (*z* in the middle of the sixth zone) for all three selected steels: for shot 1—in the upper figure, for shot 4—in the middle figure, for shot 7—in the bottom drawing.

**Figure 11.** Temperature distributions *Ti*(*t*,*r*, *z*) along the barrel thickness for selected steels for the first, fourth and seventh shots. The sign: P6.5—in the middle of the S6 zone.
