*2.4. Temperature Distibution in the Cannon Barrel for a Series of Sixty Shots*

Thermophysical properties, i.e., thermal conductivity, specific heat and density as a function of temperature in the range of 1000 ◦C to 1300 ◦C, were obtained by the linear extrapolation of experimental data of the selected steels in the RT range up to 1000 ◦C. For the selected steels, temperature distributions *Ti*(*t*,*r*, *z*) of the barrels inner surface and along the barrel thickness for a sequence of sixty shots in each zone S1 to S6 (*z* in the middle of each zone) are shown in Figure 12 for 30HN2MFA steel and Figure 13 for DUPLEX steel. The results for the 38HMJ steel are very similar to the results for the 30HN2MFA steel, therefore they are not shown in a separate drawing.

**Figure 12.** Temperature distribution *Ti*(*t*,*rw*, *z*) along the barrel thickness at the zones S1 to S6 (*z* in the middle of each zone) for the sequence of sixty shots, for the 30HN2MFA steel: black line—on the inner surface of the barrel, red line—0.1 mm below the inside surface, blue line—0.5 mm below the inside surface, green line—1 mm below the inside surface, violet line—2 mm under the inner surface, yellow line—5 mm under the inner surface, light blue line—on the outer surface of the barrel. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

**Figure 13.** Temperature distribution *Ti*(*t*,*rw*, *z*) along the barrel thickness at the zones S1 to S6 (*z* in the middle of each zone) for the sequence of sixty shots, for the DUPLEX steel: black line—on the inner surface of the barrel, red line—0.1 mm below the inside surface, blue line—0.5 mm below the inside surface, green line—1 mm below the inside surface, violet line—2 mm under the inner surface, yellow line—5 mm under the inner surface, light blue line—on the outer surface of the barrel. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

Figure 14 shows the envelopes of the so-called highest peak temperatures and lowest peak base temperatures for 60 shots for all selected steels. For 38HMJ and 30HN2MFA steels, the lines match blue.

**Figure 14.** Envelope of the lowest and the highest temperature of the inner surface at the zones S1 to S6 (*z* in the middle of each zone) for the sequence of sixty shots, for chosen steels: 30HN2MFA, 38HMJ, DUPLEX. The signs: P1.5—in the middle of the S1 zone, P2.5—in the middle of the S2 zone, etc.

#### **3. Discussion**

The results of numerical tests of the heat transfer in the barrel of the 35 mm anti-aircraft gun made of three selected steel grades showed a similar nature of heat transfer in the 38HMJ and 30HN2MFA steels, but a different one in the Duplex steel. The heat transfer model is relatively simple and the obtained values of temperature fields on the inner surface of the barrel were not overestimated, as they were in the paper [24]. The division of the cannon into six zones S1 to S6 was agreed with the Polish weapons manufacturer. As numerical tests of heat transfer in the barrel are very time-consuming, such a division method reduces the calculation time to several hours for one steel.

In the case of a single shot—the temperature on the inner surface of the barrel:

In DUPLEX steel, the maximum temperature of the inner surface of the barrel, i.e., socalled highest peak temperature was about 87 ◦C higher than for the other two steels, and it dropped less than the other two. It results from a lower value of thermal diffusivity coefficient as a function of the Duplex steel temperature in relation to the other two. In each of the three selected steels, the maximum temperature of the inner surface of the barrel occurred in the third zone, S3. In the second, S2, and fourth, S4, zones, the temperatures were not much lower. In addition, the maximum temperatures on the inner surface of the barrel in the first four zones, S1 to S4, occurred before the exit of the bullet from the barrel, while the maximum temperatures in the fifth zone, S5, and the sixth zone, S6, occurred after the bullet exited the barrel.

In the case of the sequence of seven shots—temperature on the inner surface of the barrel:

In DUPLEX steel in the S3 zone, the highest temperature and the lowest temperature of the inner surface, i.e., the so-called highest peak temperature and lowest temperature of the peak base after the seven shots reached the values of 1088 ◦C and 462 ◦C, respectively— Figure 9. The difference between the highest and lowest inner surface temperature was maximum and equal to 626 ◦C in the S3 zone and minimum and equal to 468 ◦C in the S6 zone—Figure 9. In the 38HMJ and 30HN2MFA steels in the S3 zone, the lowest and the highest temperature of the inner surface after seven shots reached the values of 345 ◦C and 1039 ◦C for 30HN2MFA steel, and 355 ◦C and 991 ◦C for 38HMJ steel, respectively. The difference between the highest and lowest inner surface temperature for these steels was maximum and equal to 694 ◦C for 30HN2MFA steel and 636 ◦C for 38HMJ steel in the S3 zone and the minimum and equal to 552 ◦C for 30HN2MFA steel and 507 ◦C for 38HMJ steel in the S6 zone. For each of the selected steels, the lowest temperature reached its maximum in the S6 zone, and the increase in the lower internal surface temperature, i.e., in the lowest temperature of the peak base became an almost linear function of the number of shots.

In the case of the sequence of seven shots—the calculation of heat transfer along the barrel thickness:

On each curve of increasing temperature of the inner surface of the barrel during the shot, we could distinguish the so-called highest peak temperature and the lowest base temperature, which was in fact the temperature of the inner wall of the barrel. This temperature was practically identical to the temperature of the barrel wall at a depth of 0.5 mm below its inner surface—Figure 10. Therefore, it can be assumed that the temperature of the inner surface of the barrel during a series of shots is equal to the barrel temperature at a depth of 0.5 mm below its surface. Temperature distributions *Ti*(*t*,*r*, *z*) along the barrel thickness for selected steels for the first, fourth and seventh shots are shown in Figure 11. At a depth of 0.5 mm, the greatest temperature difference occurred between 30HN2MFA, 38HMJ steels and DUPLEX steel. The difference increased with subsequent shots and after the seventh shot in the zone S6 it was about 86 ◦C—Figure 11 (lower drawing).

In the case of the sequence of sixty shots—the temperature along the barrel thickness:

After sixty shots, the highest internal surface temperature of the barrel occurred in the S3 zone and for all three steel grades, i.e., 30HN2MFA, 38 HMJ and DUPLEX steel, it was similar and amounted to approx. 1363 ◦C for 30HN2MFA and 38 HMJ steels, and 1348 ◦C for DUPLEX steel—Figures 12–14.

The lowest temperature of the internal surface in DUPLEX steel occurred in zones S1 and S2, i.e., around 849 ◦C in zone S1 and 916 ◦C in zone S2. For the remaining steels, the lowest temperature also occurred in zones S1 and S2, i.e., about 748 ◦C for 30HN2MFA and 38 HMJ steels in zone S1 and 827 ◦C for 30HN2MFA and 38 HMJ steels in zone S2. This is due to the fact that DUPLEX steel heats up to a higher temperature and cools down slowly as it has a lower thermal diffusivity coefficient than 30HN2MFA and 38HMJ steels. As it cools more slowly, the highest internal surface temperature after sixty shots is also lower than it would have been if this shielding effect, which is associated with an increase in inner wall temperature after each shot, was not observed in the heat flux density calculation. It should also be remembered that for 38HMJ and 30HN2MFA steels there is a shrinkage of the material and a phase change at a temperature of about 800 ◦C. The internal surface temperature of about 800 ◦C was achieved in 30HN2MFA and 38HMJ steels after about thirty shots in zone S5 (after 3.0 s) and S6 (after 2.9 s) and after about sixty shots in zone S1 (after 5.9 s)—Figure 12. It also means that the inner surface of the barrel in zones S5 and S6 will wear out the fastest. Due to the integral heat propagation effect in the steel, no changes were observed in the highest or lowest temperature of the inner surface of the barrel in the case of 30HN2MFA or 38HMJ steels specifically related to this shrinkage effect. After sixty shots, the maximum temperature of 30HN2MFA and 38HMJ steels and Duplex steel in the S2 to S4 zones was practically the same and was about 1363 ◦C for 30HN2MFA and 38 HMJ steels and 1348 ◦C for DUPLEX steel, while the lowest temperature in the 30HN2MFA and 38HMJ steels in the S5 and S6 zones was about 892◦, and in DUPLEX steel it was about 955 ◦C in zone S5 and 944 ◦C in zone S6. After sixty shots, the outer surface of the barrel in zone S6 heated up to a temperature of about 226 ◦C for 30HN2MFA steel and 230 ◦C for 38 HMJ steel, and to 103 ◦C for DUPLEX steel–Figures 12 and 13. For selected steels, the envelopes of the highest and the lowest temperatures are shown in Figure 14. There were only differences between DUPLEX steel and the other steels, i.e., 30HN2MFA and 38HMJ.

#### **4. Conclusions**

The calculations of the heat transfer in the barrel of the 35 mm anti-aircraft gun were made for the temperature-dependent thermophysical parameters, i.e., thermal conductivity, specific heat and thermal expansion (in the RT range up to 1000 ◦C) of the selected barrel steels. The paper indicates that the energy of the phase transformation should not be taken into account multiple times, e.g., both in terms of thermal conductivity and specific heat.

The results of the numerical simulation of the heat transfer in the barrel of the 35 mm anti-aircraft cannon are summarized as follows:

(1) After the first shot, the maximum temperature on the inner surface of the barrel, the so-called highest temperature is the highest in DUPLEX steel—Figure 7. The temperature difference in relation to the other two steels is about 87 ◦C in zone S3 and decreases with successive shots, and after about 4 s and about 40 shots it is similar to the temperature of DUPLEX steel (in zone S3)—Figure 14. After sixty shots, the highest temperature of the 30HN2MFA and 38HMJ steels begins to exceed the highest temperature of the DUPLEX steel by about 15 ◦C, mostly in zone S3 and S4—Figure 14;

(2) After the first and subsequent shots, when the projectile leaves the barrel, instability appears in the calculations of *Ti*(*t*,*rin*, *z*) in zones S5 and S6—Figure 8. This is related to a sharp drop in the heat transfer coefficient *hi*(*t*) in these zones, much greater than in other zones—Figure 2;

(3) In Figure 9, for a series of 7 shots, it can be seen that the shape of *Ti*(*t*,*rin*, *z*) is the same in all zones from S1 to S6—Figure 9. In all zones, the so-called lowest temperature is highest for DUPLEX steels in each zone;

(4) The so-called lowest temperature on the inner surface of the barrel *Ti*(*t*,*rin*, *z*) plays a very important role in the analysis of heat transfer in the barrel, because it is related to the phase transition of the steel from which the barrel is made. It can be assumed that the temperature of the inner surface of the barrel during a series of shots is equal to the barrel temperature at a depth of 0.5 mm below its surface, i.e., *Ti*(*t*,*r* = *rin* − 0.5 mm, *z*) —Figures 10 and 11;

(5) For the 30HN2MFA and 38HMJ steels, for which the phase transition takes place, the temperature of 800 ◦C appears in different zones at different times, the fastest in zones S5 and S6. This means that zones S5 and S6 of the barrel will be exposed to the greatest wear. Already after about 3 s, i.e., after about thirty shots, these parts of the barrel will exceed the phase transition temperature—Figure 12;

(6) The so-called lowest temperature on the inner surface of the barrel *Ti*(*t*,*rin*, *z*) made of DUPLEX steel is always higher than the same temperature for a barrel made of 30HN2MFA or 38HMJ steel—Figure 14. In 2 s, i.e., after twenty-five shots, the difference between them in zones S2 and S3 is greatest at about 136 ◦C, after sixty shots it will drop in zone S6 to about 52 ◦C;

(7) Due to the lack of a phase transition, the DUPLEX steel can operate above the temperature of 800 ◦C. This steel does not have the material shrinkage effect and therefore repeatedly exceeding this temperature in the process of heating and cooling the barrel has no effect on the formation of cracks on the inner surface of the barrel.

**Author Contributions:** Conceptualization, M.Z., P.K. and Z.S.; methodology, P.K., M.Z. and Z.S.; software, M.Z., Z.S.; validation, J.Z. and M.P.; formal analysis, P.K., M.Z. and Z.S.; writing—original draft preparation, P.K., M.Z. and Z.S.; writing—review and editing, P.K., M.Z. and J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research results reported in this work were obtained thanks to funding from the Polish National Centre for Research and Development 2012–2016 Scientific Fund, Project no. O ROB 0046 03 001.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
