*1.1. Determination of the Heat Transfer Coefficient and Gas Temperature*

The heat transfer coefficient can be calculated from a convective heat transfer correlation for a fully developed turbulent pipe flow, which expresses the Nusselt number *Nu* as a function of the Reynolds *Re* and Prandtl *Pr* numbers [25]:

$$Nu\_D = 0.023 \, Re\_D^{0.8} Pr^{0.4} \tag{1}$$

For fluid flow in a pipe of circular cross section of diameter *D*, if the gas has a velocity *w*, density *ρ*, dynamic viscosity *μ* and thermal conductivity *kp*, the definitions of the Nusselt and Reynolds numbers are, respectively:

$$Nu\_D = \frac{hD}{k\_p} \text{ } Re\_D = \frac{\rho wD}{\mu} \tag{2}$$

By substituting Equation (2) into Equation (1), one can express *h* in the form:

$$h = \frac{0.023}{D^{0.2}} \cdot \frac{k\_p}{\mu^{0.8}} Pr^{0.4} (\rho w)^{0.8} \tag{3}$$

The approximate value of the Prandtl number can be calculated using the simplified Eucken formula [23,25]:

$$Pr = \frac{4\chi}{9\chi - 5} \tag{4}$$

where <sup>γ</sup> is adiabatic index of the gunpowder gases. For <sup>γ</sup> = 1.20, we have *Pr*0.4 ≈ 0.93.

Considering the weak temperature dependence of the *kp <sup>μ</sup>*0.8 ≈ 285 relation (for the average temperature of the gunpowder gases equal to 1000 ◦C), we have [25,26]:

$$h = \frac{6.1}{D^{0.2}} \left(\rho w\right)^{0.8}.\tag{5}$$

Since the density *ρ* and velocity *w* of the gunpowder gases are functions of time, we have different values of the time-dependent heat transfer coefficient *h* in the cross-section P1 to P6 of the 35 mm cannon barrel. In the adopted model of heat transfer in the gun barrel, we assume that the calculations *ρ* and *w* in the cross-section P1 to P6 are also valid in zones S1 to S6, respectively.

Density *ρ*, velocity *w* and bore gas temperature as a function of time *Tg*(*t*) can be determined by solving the interior ballistic model, which is a model with lumped parameters [27–29]. The calculations took into account the phenomena occurring in the barrel until the projectile exits the barrel and the after muzzle period. The condition for completing the calculation is that the propellant pressure in the barrel drops to 0.18 MPa [30].

When the projectile is in the barrel, the model contains:

Energy conservation equation (the first law of thermodynamics):

$$d\mathcal{U} = d\mathcal{Q} - dW \tag{6}$$

Here, differential of internal energy *dU* taking into account mass fraction of burning propellant '*zp*' with respect to its initial mass *mp* has the form:

$$d\mathcal{U} = d\left(c\_v m\_p z\_p T\right) = c\_v m\_p \left(T dz\_p + z\_p dT\right) \tag{7}$$

Amount of heat *dQ* release during burning of propellant of isochoric flame temperature *T*<sup>1</sup> and specific heat at constant volume *cv* equals:

$$dQ = c\_v T\_1 m\_p dz\_p \tag{8}$$

Amount of sum of works of propellant gases *dW* taking into account coefficient of secondary works *ϕ* is given by:

$$dW = d\left(\varphi \frac{mv^2}{2}\right) = \varphi mvdv \tag{9}$$

Substituting Equations (7)–(9) into Equation (6) and replacing *cv* by *cv* = *<sup>R</sup> <sup>γ</sup>*−<sup>1</sup> after some algebraic manipulations, we obtain:

$$\frac{d(RT)}{dt} = \frac{(f - RT)m\_p \frac{dz\_p}{dt} - (\gamma - 1) \, qmv \frac{dv}{dt}}{m\_p z\_p} \tag{10}$$

where *f* = *RT*1, *γ* = *cp*/*cv*.

Equation of state of propellant gases [27–29]:

$$p\left(V\_0 + sl - \frac{m\_p}{\rho\_p}(1 - z\_p) - \eta m\_p z\_p\right) = m\_p z\_p RT \tag{11}$$

Equation of mass fraction burning rate of the propellant (gas inflow) [27–29]:

$$\frac{dz\_p}{dt} = \frac{S\_1}{\Lambda\_1} \sqrt{1 + 4 \frac{\lambda\_1}{\kappa\_1} z\_p} \cdot r\_1 p \tag{12}$$

Equation of the projectile motion:

$$
\rho m \frac{dv}{dt} = ps\tag{13}
$$

where *ϕ* = *K* + <sup>1</sup> 3 *mp m* . Definition of the projectile velocity:

$$\frac{dl}{dt} = v\tag{14}$$

Propellant gas density:

$$\rho = \frac{m\_p z\_p}{V\_0 + sl - \frac{m\_p}{\rho\_p} \left(1 - z\_p\right) - \eta m\_p z\_p} \tag{15}$$

Assuming a linear distribution of the velocity of the propellant gases in the barrel, we calculate the velocity *w* of gases in the considered cross-section *i* = 1, ... , 6, (*i*–a cross-section number from P1 to P6) of the barrel according to:

$$w = \frac{l\_i}{l\_0 + l} \cdot v \tag{16}$$

where: *li*—distance from the bottom of the chamber to the cross-section *i* of the barrel; *l*0—length of the canon chamber, *l*—projectile travel inside the barrel.

In the period after the projectile muzzle, the model includes:

Energy conservation equation, taking into account the outflow of gases to the environment (through the muzzle of the barrel):

$$d\mathcal{U} = d\mathcal{Q} - dH \tag{17}$$

Considering that:

$$d\mathcal{U} = d\left[\mathbf{c}\_v m\_p \left(\mathbf{z}\_p - \boldsymbol{\zeta}\right)T\right] = \frac{m\_p R}{\gamma - 1} \left[T\left(d\boldsymbol{z}\_p - d\boldsymbol{\zeta}\right) + \left(\mathbf{z}\_p - \boldsymbol{\zeta}\right)dT\right] \tag{18}$$

$$dQ = c\_v T\_1 m\_p dz\_p = \frac{1}{\gamma - 1} f m\_p dz\_p \tag{19}$$

$$dH = c\_p m\_p T d\zeta = \frac{\gamma}{\gamma - 1} m\_p RT d\zeta\_{\prime} \tag{20}$$

Equation (17) takes the form:

$$\frac{d(RT)}{dt} = \frac{(f - RT)\frac{dz\_p}{dt} - (\gamma - 1)RT\frac{d\mathbb{Z}}{dt}}{z\_P - \mathbb{Z}}\tag{21}$$

Equation of state of propellant gases [27–29]:

$$p\left(V\_0 + sl\_m - \frac{m\_p}{\rho\_p}(1 - z\_p) - \eta m\_p (z\_p - \zeta)\right) = m\_p (z\_p - \zeta)RT\tag{22}$$

Propellant gas density:

$$\rho = \frac{m\_p \left(z\_p - \zeta\right)}{V\_0 + s l\_m - \frac{m\_p}{\rho\_p} \left(1 - z\_p\right) - \eta m\_p \left(z\_p - \zeta\right)}\tag{23}$$

Rate of mass fraction of propellant gases flowing out of from the barrel (gas outflow):

$$\frac{d\overline{\zeta}}{dt} = \frac{sp}{m\_p \sqrt{RT}} \sqrt{\gamma \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma + 1}{\gamma - 1}}} \tag{24}$$

Assuming that the propellant gases flowing out of the barrel reach critical parameters, their velocity in the considered cross-section *i* of the barrel will be calculated according to:

$$w = \frac{l\_i}{l\_0 + l\_m} \cdot w\_{cr} \tag{25}$$

where *wcr* <sup>=</sup> <sup>√</sup>γ*RTcr* <sup>=</sup> <sup>2</sup><sup>γ</sup> <sup>γ</sup>+1*RT*; *wcr*, *Tcr*—critical velocity and critical temperature of propellant gases at the muzzle.

The initial conditions for calculations are the following:

*t* = 0, *RT* = *RT*<sup>1</sup> = *f* , *zp* = 0.001, *l* = 0, *v* = 0, *ζ* = 0.

It should be added that the variables *zp(t)*, *l(t), v(t), T(t), ζ*(*t*)*, w*(*t*), *ρ*(*t*) and *p(t)* are functions of time. The input data for the interior ballistic calculations are shown in Table 1.


**Table 1.** Input data to interior ballistics calculation.

The calculation results of the heat transfer coefficient as a function of time *hi*(*t*) in the six cross-sections P1 to P6 and the gas temperature as a function of time *Tg*(*t*) for the 35 mm anti-aircraft cannon barrel are shown in Figure 2, and so the values of *hi*(*t*) in the section P1 are valid in the zone S0 and S1, P2 in the zone S2, P3 in the zone S3, etc. In this paper, zone S1 includes the zone S0 of the cannon breech. Detailed and precise calculations of heat transfer in the S0 zone are not the subject of the study.

For *t* = 4.54 ms, we observe rapid drops of the heat transfer coefficient as a function of time *hi*(*t*) in each of the six cross-sections P1 to P6. This is the moment when the bullet leaves the barrel. The highest value is achieved by the heat transfer coefficient in the fourth zone. It has a slightly lower value in zone three.
