**1. Introduction**

It is well known that modern anti-aircraft artillery systems consist of a number of guns, some of which fire at the designated target, while the remaining guns follow the target without firing a shot. This is due to the timing of the single cannon being fired, which is chosen because the high temperatures in the barrel prevent the gun from being fired. In the case of overheating the barrel of one of the guns, it loses its ability to fire. Shooting is then taken over by the other guns that track the target on standby mode. It is also possible to fire all battery guns at once. Low barrel life is a bottleneck that limits the improvement of the weapon's performance for a long time. Many years of research have shown that the erosion of the internal bore is a direct cause that affects the service life of the barrel. The erosion of the inner surface of the barrel is caused by the action of heat, chemistry and mechanics, with heat playing a leading role [1–8]. Although the mechanism of gun barrel wear is not fully understood, it is known that wear is very closely related to the maximum temperature of the bore surface [9]. Usually, when designing the firing cycle, it is essential to maintain the temperature below 800 ◦C, established by the manufacturer as a maximum temperature when testing the gun barrel's life [9,10]. The maximum temperature limit of the barrel bore in operation is dictated by the thermophysical properties of the steel grade of the barrel. In the steel grades under consideration in our paper, a temperature above 800 ◦C causes allotropic changes connected with the reconstruction of the crystal lattice of the alloy [11,12]. The kinetics of change is well described by the dilatometric

**Citation:** Zieli ´nski, M.; Koniorczyk, P.; Surma, Z.; Zmywaczyk, J.; Preiskorn, M. Numerical Study of Heat Transfer in a Gun Barrel Made of Selected Steels. *Energies* **2022**, *15*, 1868. https://doi.org/10.3390/ en15051868

Academic Editor: Dmitry Eskin

Received: 13 January 2022 Accepted: 25 February 2022 Published: 3 March 2022

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curve characteristics for the individual steel grades [11–13]. The effect of temperature on the barrel bore alters the volume of the surface layer, giving rise to a typical mesh of cracks. This affects the flaking off of the protective coating on the inner surface of the barrel. In older technological processes, the protective coating consists of electroplated chromium. Currently, this process is being replaced by nitriding. In any case, the protective coating is corroded by the structural transformations in the steel layer, which are directly related to the phase transition between ferrite and austenite [3,12,14]. At present, research is being conducted on the implementation of new steel grades with a higher allotropic transition temperature into the production of gun barrels [12]. It is about shifting the ferrite–austenite phase transition towards a higher temperature or using steels in which this transition does not take place [11]. When calculating the heat transfer in the barrel, constant values of thermal conductivity, specific heat and the density of the barrel material are often taken [9,14,15]. Many publications believe that the thermal properties of the gun barrel material are temperature dependent [16–20]. As a rule, the temperature dependence of the barrel material density is neglected due to small changes [16]. It is very important to correctly introduce the thermophysical properties of new steel grades as input data for the heat transfer calculations in the barrel. We consider a phase change only in relation to the thermal conductivity. However, in the literature, one can find papers in which the thermal effect of the phase transformation has been included twice, i.e., in thermal conductivity and specific heat, which seems to be an erroneous [16]. Thermal diffusivity *a*, thermal conductivity *k*, specific heat *cp* and density *ρ* are related to the expression *a = k/(ρ*·*cp)*. Each of these thermophysical parameters can be determined on separate measuring stations or, for example, the thermal conductivity can be calculated from the expression *k=a*·*ρ*·*cp*. The phase transformation is visible in each thermophysical parameter. Thus, when calculating the thermal conductivity *k* in the phase transition region from formula *k=a*·*ρ*·*cp*, this effect will be taken into account both in thermal diffusivity and in specific heat. This means that the phase change effect and the associated enthalpy will be accounted for twice. As a rule, we consider the phase transition effect in thermal conductivity characteristic [21]. During the continuous firing of artillery, the inner wall of the barrel will experience a continuous rise of temperature. On each curve of the barrel temperature increase during the shot we can distinguish the so-called highest peak temperature and lowest temperature of the peak base, which is in fact the inner wall temperature of the barrel. In order to reach the temperature of 800 ◦C of the inner barrel surface, it is often necessary to carry out numerical simulations of the heat transfer in the barrel after firing several dozen shots [14,18,19].

Over the years, numerous research groups have carried out a series of tests to determine the temperature field of the gun barrel. These calculations are becoming more and more accurate and verifiable in experimental research [1,15,22,23]. However, it is often important to simulate heat transfer throughout the barrel, not just a fragment of it. In order to avoid very time-consuming calculations, the barrel can be divided into sectors. In this paper an initial boundary value problem (IBVP) of heat transfer in the barrel wall of a 35 mm caliber cannon was solved for the single shot and the sequence of seven shots for chosen barrel steels. For calculation purposes, the barrel with a total length of 3150 mm has been divided into six zones S1 to S6—Figure 1. The heat transfer coefficient was calculated as a function of the time *hi*(*t*) in the six cross-sections P1 to P6 on the inner surface of the barrel: P1: z = 216 mm, P2: z = 385 mm, P3: z = 535 mm, P4: z = 880 mm, P5: z = 2081 mm, P6: z=2980 mm and the gas temperature as a function of time *Tg*(*t*). The functions *hi*(*t*) in cross-sections P1 to P6 are valid in the zones S1 to S6. Additionally, the S0 zone of the cannon breech was distinguished in the range from 0 to 216 mm, for which—at the present stage of the research—the same function *hi*(*t*) was assigned as for the S1 zone. The calculations were carried out considering the temperature-dependent thermophysical parameters in the model, i.e., thermal conductivity, specific heat and thermal expansion (in the range from RT to 1000 ◦C). In 2020, the authors of this study tested the thermophysical properties of selected barrel steels, i.e., 38HMJ (1.8509), 30HN2MFA and DUPLEX (1.4462) (in the range from RT to 1000 ◦C) [12]. In this study, particular attention was paid to the correct

introduction of thermophysical parameters depending on the temperature in the numerical heat transfer tests in the barrel wall of a 35 mm caliber cannon for a single shot and the sequence of shots for the chosen barrel steels. The idea is not to erroneously consider the phase transition effects on the selected metals twice, such as in thermal conductivity and specific heat.

**Figure 1.** Heat transfer zones S1 to S6 of the 35 mm cannon barrel input to the calculations: S1: 0 ÷ 385 mm, *rout* = 55.0 ÷ 55.0 mm; S2: 385 ÷ 535 mm, *rout* = 55.0 ÷ 57.0 mm; S3: 535 ÷ 880 mm, *rout* = 57.0 ÷ 59.5 mm; S4: 880 ÷ 2081 mm, *rout* = 59.5 ÷ 44.07 mm; S5: 2081 ÷ 2980 mm, *rout* = 44.07 ÷ 31.0 mm; S6: 2980 ÷ 3150 mm, *rout* = 31.0 mm. The zone S1 includes the zone S0 of the cannon breech (reproduced with permission from [24], Military University of Technology, 2022).
