Modeling of convective heat transfer processes between inhomogeneous gas mixtures and surfaces of a small-caliber artillery barrel

Texto integral

Introduction

The phenomenon of a shot, the duration of which is measured in thousandths of a second, is inevitably associated with the action of high temperature gases on the bore and heat removal from the outer wall of the barrel. While studying the functioning of AAW in a thermal setting, as a rule, the boundary layer theory known in thermodynamics is used [1; 2], according to which the temperature loads on the object under study depend only on the instantaneous values of the coolant temperature, determined from the energy characteristics of the heat source. This hypothesis of a brief history does not consider the change in the thermophysical characteristics of individual gas components during a shot (thermal conductivity coefficients λ₁, dynamic viscosity η₁, specific heat capacity c₁) and it does not take into account the unsteady effects associated with the dependence of the physical parameters of the gas mixture on the current temperature of the gases T₁. One of the ways to consider changes in the physical coefficients of the gas mixture as combustion and charge advances is to use provisions on the non-stationary parameters of gas flow [3; 4].

There is also heat transfer from the heated sections of the barrel to the environment (air) during a shot. In general, heat exchange between gases and the barrel, the outer surface of the barrel and air can occur through forced convection and radiation. Comparing to long thin-walled barrels for tank and anti-tank guns, solar radiation has a significantly smaller effect on the temperature field of AAW barrels than convection [5]. The influence of radiation on the formation of the barrel temperature T is not studied by this research.

Therefore, the entire complexity of studying the physical essence of the transformation of gunpowder energy into the energy of directed transfer in time and space of a projectile, accompanied by factors affecting the barrel, is connected with the determination of the value of the total heat transfer coefficient α_Ι, Ι = 1,2, taking into account heat transfer processes from gases to the barrel (Ι = 1) and from the outer surface of the barrel to the air (Ι = 2).

Calculating the state parameters of gas and air flows is a very complex gas-dynamic problem. The previously published article [6] presents the mathematical reproduction of heat transfer on the walls of the barrel as an independent problem. The investigations show [2; 7], that the intensity of heat exchange between the surface of the body and the coolants in one way or another functionally depends on many parameters: the geometric shape and size of the body; physical properties, direction and speed of coolant flow; temperature conditions of interaction between body and substance, and others. As a result, the research highlights as priorities the issues of justifying a method of formalizing the processes of heat transfer from a turbulent gas flow to the channel wall and from the outer wall of the barrel into the atmosphere, acceptable for engineering calculations.

In analytical studies, the heat transfer process is described by a system of differential equations that takes into account both thermal and hydrodynamic phenomena and includes the equations of heat transfer, heat transfer, motion, and continuity [8–10]. It is difficult to obtain an exact solution to the problem in analytical or numerical form, even with high-performance computers, since the longitudinal flow around a pipe of circular cross-section is characterized by both strong unevenness of gas state parameters and turbulence, and a very complex geometry of the computational domain. Therefore, to calculate such flows, the approaches related to averaging the equations of motion and further consideration of the flow in the form of a continuous flow have become relatively widespread. An example of this approach is, for example, the paper [11], where calculating the flow is based on a mathematical model of a porous isotropic body. As this approximate approach is developed, various options to derive the averaged transport equations used are proposed, and various ways to take into account flow turbulence and other flow features are considered. A slightly different approximate method for solving the problem is also known [12], when the real flow through the pipe is replaced by a continuous “homogeneous” flow without involving a mathematical model of a porous body. In principle, both the first and second approaches allow to obtain information about the dynamic and thermal characteristics of the phenomenon averaged over the volume of the body. In this case, using empirical initial data and assumptions have to be used, they are in a relatively good agreement with the real structure of the flow. It is explained, first of all, by the fact that the data necessary to solve the problem can be found only approximately and in some cases with a large error. This approximate assessment of the initial data is caused by the lack of uniform dynamic dependencies.

The presented above approaches should be noted not to have lost their relevance yet. Therefore, the publication [13] argues a unique version of the thermal model developed on the basis of the apparatus of probability theory; the articles [14; 15] propose schemes for experimental studies and methods for processing output data that provide increased accuracy in determining body temperature; the investigation [16] determines the temperature fields of finned walls of various configurations with numerical solutions of the multidimensional heat conduction problem; the research [17] proposes tools for modelling the temperature field in gas turbine assemblies, taking into account as much as possible the set of parameters in multifactor boundary conditions of the boundary layer. The publications [18–22] present the examples of exploration on similar topics in the field of aviation artillery science.

The methods developed in these areas are completely objective and can serve as the basis for calculating the heat transfer characteristics on the surface of a pipe of complex geometric shape.

 

Justification and specification of the criterion equations for heat transfer of the phenomenon of a shot from AAW

A feature of heat transfer from hot gases to the barrel bore and from the outer surface of the barrel to the air is that the medium where heat spreads moves. The movement results in transferring heat along with the gas mass. It is obvious that the nature of heat transfer is determined by the shape of the region and the properties of interaction flows - velocity distribution, flow regime, and others.

Based on the above, the approach of calculating the total heat transfer coefficient α_Ι, Ι = 1,2, using statistical formulas where there are the criterion dependencies of the similarity theory - Reynolds number Re, Nusselt number Nu, Prandtl number Pr [3; 4; 8; 9].

Since the desired value α_Ι, Ι = 1,2, is total or effective, the amount of heat transferred will be the sum of two independent terms of heat transferred by forced convection from the gases to the barrel bore and from the outer wall of the barrel to the air. Both of these terms can be calculated separately and then added together.

Then

${\text{α}}_{{\rm I}}=\frac{{\text{Nu}}_{{\rm I}}{\text{\hspace{0.17em}λ}}_{{\rm I}}}{l},{\rm I}=\mathrm{1,2}.$ (1)

To calculate by formula (1) the number Nu_Ι, Ι = 1,2 – a dimensionless quantity characterizing the intensity of heat exchange along the length of the barrel l at its internal (Ι = 1) and external (Ι = 2) boundaries, the researchers used the dependencies where the similarity criteria were constructed according to the defining dimensions of the space where heat transfer occurred, and the flow temperature along the length of the barrel l was chosen as the determining one: gas temperature T₁ inside, air temperature T₂ outside the barrel. The gas temperature T₁ is calculated by solving the main problem of internal ballistics [23]. An array of air temperature values T₂ as a function of the altitude of AAW application is specified based on the standard atmosphere SA-81 [24]. The designations of the thermal conductivity coefficients λ_I, Ι = 1,2, similar to the designation of the classifier of heat transfer boundaries in the barrel, have the meaning of the thermal conductivity of gases λ₁ (Ι = 1) and air λ₂ (Ι = 2).

When modelling the movement of a coolant in the barrel and air during longitudinal flow around the outer surface of the barrel at Mach number M > 1, a turbulent boundary layer is observed, that is a turbulent flow regime is realized (Re₁ 10⁴), and with the ratio of the defining dimensions (length l and diameter d ) of the barrel bore l/d $\approx$ 50, heat transfer can be calculated using formulae for characteristics of the coolant flow regimes.

It is obvious that convective heat exchange in the barrel is of an unstabilized nature of a turbulent flow regime, since the flow of gases occurs in a small volume. Some stabilization of the flow is observed in case when the moving boundary of interaction (the bottom of the projectile) moves towards the muzzle. The general form of the criterion equation for gases during coolant flow in channels of annular cross-section under the specified conditions (Re₁ > 10⁴) has got the form:

${\text{Nu}}_1={\rm A}{\mathrm{Re}}_1^{\text{κ}}{\mathrm{Pr}}_1^{\text{υ}}{k}_z,$ (2)

where Α, κ, υ – coefficients determined by experiments; k_z – correction factor taking into account the change in the heat transfer coefficient from gases to the barrel bore α₁ in the flow stabilization section.

Due to generalizing the existing data [3; 25–28], the following relationship is established to determine the number Nu₁ in expression (2) for forced convection of gases in a finned channel with an annular cross-section:

 

${\text{Nu}}_1=\mathrm{0,023}{\mathrm{Re}}_1^{\text{0,8}}{\mathrm{Pr}}_1^{\text{0,43}}.$ (3)

Formula (3) obtains the coefficient k_z = 1 with l $\ge$15d.

Heat will be removed from the outer surface of the barrel by an air flow with high blowing speeds. Since the air flow occurs in a much larger volume, convective heat exchange at the outer wall of the barrel is subject to the transitional flow regime and it is characterized by greater stabilization of the flow.

During a transitional (2100 < Re₂ 10⁴) stabilized air flow, the following relationship could determine the Nu₂ number:

 

${\text{Nu}}_2={\rm A}{\mathrm{Re}}_2^{\text{κ}}.$ (4)

By processing the most accurate data, several authors have obtained an expression to determine the number Nu₂ when air flows along the surface of a cylinder of variable cross-section [9; 25; 29; 30]. Processing of experimental data has showed that the values of the number Nu₂ in expression (4) are proportional to the number :

${\text{Nu}}_2=\mathrm{0,034}{\mathrm{Re}}_2^{\text{0,8}}.$(5)

Formula (5) is valid when the ratio of the internal and external radii of the cylinder does not exceed 0.2. Due to the researchers’ report, the experimental data are in good agreement within 10–12%.

The numbers Re_I, Ι = 1,2, in formulae (2), (4), characterizing the ratio of inertial forces to molecular friction forces, are determined according to the following expressions:

– Re₁ number in formula (2) for gases flowing in the barrel channel could be placed in the functional connection with the velocity of the gases during the shot v₁, which is also calculated by solving the main problem of internal ballistics [23] and the reference value of the coefficient of dynamic viscosity of gases η1 [3; 4]:

 

${\text{Re}}_{\text{1}}=\frac{v_{1}l}{{\text{η}}_1};$ (6)

– Re₂ number in formula (4) for air flowing along the surface of the barrel will be calculated based on the value of the speed of the oncoming air flow during firing v₂, which, with a comfortable (intra-fuselage) placement of the AAW, is identified with the speed of the aircraft and also the nominal value of the coefficient of air kinematic viscosity μ₂ [25]:

 

${\text{Re}}_{\text{2}}=\frac{v_{2}l}{{\text{μ}}_2}.$

The number Pr1 in formula (2) characterizes the similarity of speed and temperature fields in the gas flow, it is determined by the formula

${\text{Pr}}_{\text{1}}=\frac{v_1}{a_1}=\frac{{\text{η}}_1{c}_1}{{\text{λ}}_1},$ (7)

where a₁ – temperature conductivity coefficient of gases.

In gas-pulse systems, which include AAW samples, large specific heat fluxes are often transmitted. Intensive heat exchange of gases with the surfaces of the barrel is achieved at significant temperature gradients of the heat exchanger along the barrel bore radius, that is at large temperature differences
[T₁ – T], [T – T₂]. The thermophysical properties of coolants depend, first of all, on the instantaneous values of the gas temperature T₁, and to calculate the physical characteristics of gases, it is necessary to know the physical characteristics of their components at high temperatures. Therefore, to calculate the values of the defining criteria of similarity and heat transfer coefficient, it is necessary to calculate the values of the physical parameters of gases that are a gas mixture.

 

Method for taking into account the dependence of the physical parameters of a gas mixture on the current gas temperature

In the planned specifications in modeling convective processes of near-wall flows, we will follow the fact that the gunpowders used to equip unitary cartridges for AAW, in terms of their chemical elemental composition, are compounds of carbon, hydrogen, oxygen, nitrogen with an intramolecular oxygen content sufficient for the complete conversion of combustible elements into gas. Therefore, their burning results mainly in obtaining gaseous products, and only in some cases a small amount of solids is formed.

Research shows [4; 31], that the main combustion products of gunpowder are carbon dioxide CO₂, carbon monoxide CO, nitrogen N₂, hydrogen H₂ and water vapor H₂O. In some cases, the transformation products may contain CH₄ methane, but under normal combustion conditions these products are contained in very small quantities.

The composition of the transformation products depends on the nature of the gunpowder and the conditions under which the gunpowder of a given nature burns. The greater the oxygen balance of gunpowder, the more combustion products contain carbon dioxide CO₂ and water H₂O, that is, products of complete oxidation. The lower the oxygen balance of gunpowder, the more products of incomplete combustion – carbon monoxide CO and hydrogen H₂. The ratio between the main products is determined by the equilibrium of the water gas reaction:

СО + Н2О ↔ СО2 + Н2.

 

The qualitative and quantitative composition of combustion products may vary somewhat depending on the gas pressure during the shot and the cooling conditions of the products, since when gunpowder burns in a closed volume, the gas pressure during the shot is determined by the loading conditions of the AAW. Increasing loading conditions and increasing gas pressure during firing, the content of carbon dioxide CO₂ and methane CH₄ increases slightly, and the content of carbon monoxide CO and hydrogen H₂ decreases. This is explained, on the one hand, by the course of secondary reactions, and on the other, by a different direction of explosive transformation reactions. The appearance of nitrogen oxides NO in the combustion products of gunpowder is due to the fact that combustion occurs in several stages, described by the Serebryakov – Zeldovich – Belyaev temperature profile [1; 4; 31]. Nitrogen oxides NO are intermediate products and in the last stage of gunpowder combustion they interact with the products of incomplete combustion according to the reactions:

 

2CO + 2NО ↔ 2CО2 + N2+ q;              2H2 + 2NО ↔ 2H2О + N2+q,

 

where q – heat from gas combustion reaction.

The theoretical analysis of the chemical composition, percentage contents and energy characteristics of gunpowders used to equip unitary cartridges for AAW demonstrated that to determine the coefficients of thermal conductivity, dynamic viscosity and heat capacity of the ι-th component of the gas mixture (${\text{λ}}_{1_{\iota }},{\text{η}}_{1_{\iota }},{с}_{1_{\iota }},\iota =\overline{\mathrm{1,5}}$), approximating analytical dependencies can be used [1; 25], the dependencies specify formulae (6), (7).

For gas СО₂ (ι = 1):

${\text{λ}}_{1_1}=(5,89832+0,0963058T_1-0,000757319T_1^{3/2})\cdot {10}^{-3};$ (8)

${\text{η}}_{1_1}=(\mathrm{5,60846}+0,043375T_1-0,0007347768T_1^{3/2})\cdot {10}^{-3};$ (9)

${с}_{1_1}=\mathrm{21,6966}+0,090772T_1-0,00243224T_1^{3/2}+\mathrm{0,0000187612}{T}_1^2.$ (10)

For gas СО (ι = 2):

 

${\text{λ}}_{1_2}=(\mathrm{4,01627}+0,0861T_1-0,000614918T_1^{3/2})\cdot {10}^{-3};$                      (11)

${\text{η}}_{1_2}=(\mathrm{8,19784}+0,043251T_1-0,000325791T_1^{3/2})\cdot {10}^{-3};$                    (12)

${с}_{1_2}=\mathrm{24,4476}+0,0181285T_1-0,000357338T_1^{3/2}-\mathrm{1,89466}\cdot {10}^{-6}{T}_1^2;$ (13)

 

For gas H₂O (ι = 3):

 

${\text{λ}}_{1_3}=(-\mathrm{20,3507}+\mathrm{0,17663}{T}_1-0,00057079T_1^{3/2})\cdot {10}^{-3};$                    (14)

${\text{η}}_{1_3}=(-\mathrm{0,0346853}+\mathrm{0,55828}{T}_1-0,000452251T_1^{3/2})\cdot {10}^{-6};$                (15)

${с}_{1_3}=\mathrm{28,90216}+0,0105406T_1+0,000198288T_1^{3/2}-\mathrm{4,36359}\cdot {10}^{-6}{T}_1^2.$(16)

 

For gas H₂ (ι = 4):

${\text{λ}}_{1_4}=(\mathrm{80,3534}+\mathrm{0,375747}{T}_1-0,000917275T_1^{3/2})\cdot {10}^{-3};$                   (17)

${\text{η}}_{1_4}=(\mathrm{4,28174}+\mathrm{0,019866}{T}_1-0,000146665T_1^{3/2})\cdot {10}^{-6};$                     (18)

${с}_{1_4}=\mathrm{30,9404}-0,0147268T_1+0,000637427T_1^{3/2}-\mathrm{6,05625}\cdot {10}^{-6}{T}_1^2.$ (19)

 

For gas N₂ (ι = 5):

 

${\text{λ}}_{1_5}=(\mathrm{5,1584}+\mathrm{0,0784756}{T}_1-0,000534648T_1^{3/2})\cdot {10}^{-3};$                    (20)

${\text{η}}_{1_5}=(\mathrm{8,47204}+\mathrm{0,0401594}{T}_1-0,000297397T_1^{3/2})\cdot {10}^{-3};$                    (21)

${с}_{1_5}=\mathrm{25,2938}+0,0128032T_1-0,000180876T_1^{3/2}+\mathrm{3,33147}\cdot {10}^{-7}{T}_1^2.$ (22)

 

Formulae (8)–(22) are applicable at gas temperatures of 400 K ≤ T₁ ≤ 3000 K.

Calculating the physical characteristics of a mixture from the values for individual gas components can use methods developed for studying objects in related subject areas and systematized in the source [25].

To calculate the thermal conductivity coefficient of a mixture of gases, the dependence is applicable

 

$\sum _{\iota }^5{\text{ε}}_{\iota }\cdot m_{\iota }\cdot {\text{λ}}_{1_{\iota }}$ (23)

where ε_ι – mole fraction of the ι-th component of a gas mixture; m_ι – mole mass of the ι-th component of a gas mixture; m_sm – mole mass of a gas mixture.

The values of mole fractions ε_ι and molar masses m_ι of ι-ths components of a mixture of gases ($\iota =\overline{\mathrm{1,5}}$) are available in reference books, problem books or applications to the textbooks, for example, [3; 4; 25]. To calculate the molar mass of a mixture of gases in expression (23), the formula known in thermophysics is used:

 

$m_{\text{см}}=\sum _{\iota }^5{\text{ε}}_{\iota }\cdot m_{\iota }.$

To calculate the coefficient of dynamic viscosity of a gas mixture, there is formulation

${\text{η}}_1={\left(\sum _{\iota }^5\frac{{\text{ε}}_{\iota }}{{\text{η}}_{1_{\iota }}}\right)}^{-1}.$

Calculating the coefficient of heat capacity of a mixture of gases is carried out according to the expression

${с}_1=\sum _{\iota }^5{\text{ε}}_{\iota }\cdot {с}_{1_{\iota }}.$

It should be noted that the given criterion equations, which serve to determine the total heat transfer coefficient from gases to the barrel bore and from the outer surface of the barrel to the air α_Ι, Ι = 1,2, are not universal dependencies and they are given for very specific conditions, characterized by the shape of the cross-section of the barrel, thermophysical properties of gases and barrel steel material, as well as unsteadiness of gas flow.

When using AAW, determining the state of coolants is reduced to repeated calculation of the criterion equation (3) using formulae (6), (7) and constructing the functions:

 ${\lambda }_{1_{\iota }}\left(T_1\right),\iota =\overline{\mathrm{1,5}}$of type (8), (11), (14), (17), (20); ${\eta }_{1_{\iota }}\left(T_1\right),\iota =\overline{\mathrm{1,5}}$ of type (9), (12), (15), (18), (21); ${с}_{1_{\iota }}\left(T_1\right),\iota =\overline{\mathrm{1,5}}$ of type (10), (13), (16), (19), (22). Programme requests for gas temperature values T₁ are carried out for each step of integration of the system of internal ballistics equations grouped in [23].

The calculation of criterion equation (5) is constant at constant values of the specified speed of the aircraft v₂ and the reference value of the coefficient of kinematic viscosity of air μ2.

The completed form of modelling the processes of near-wall flows of gases and air during a small-caliber artillery shot was a computer program that allows to calculate heat transfer parameters from the values of the physical parameters of the gas mixture. The computer program was created and debugged using the Microsoft Developer Studio software product, the Fortran Power Station 4.0 environment and the Fortran-90 algorithmic language.

Due to the required level of abstraction and due to the inevitable loss of some information, the presented model does not provide a complete picture characterizing the researched physical processes. Justification of particular formulations and further discussions of the consequences seem possible after checking the adequacy of the model to real processes of heat exchange between the barrel and surrounding coolants.

 

Verification of the model adequacy

Establishing a set of properties of a model that reveals its suitability for solving a given problem is possible by comparing the results of mathematical modeling with data from natural experiments and previously obtained results of theoretical work. This approach can significantly increase the objectivity of conclusions.

Fig. 1 presents comparative graphs of the results of mathematical modeling of the heat transfer process during a shot with previously obtained illogical theoretical solutions and borrowed experimental data [32]. The adequacy of the model was verified based on mandatory compliance with the conditions of thermal and geometric similarity stated above.

Analysis of the results of calculating the intensity of the influence of coolants on the barrel demonstrated an increase in accuracy in the convergence of theoretical results of determining the parameters of heat transfer during a shot to the experiment results by 19 %. Consequently, applying the proposed tool, it is possible to more accurately determine the amount of heat entering the bore and removed from the outer wall of the barrel under various conditions of firing from the AAW.

 

Рис. 1. Зависимость полного коэффициента теплоотдачи от времени при малокалиберном артиллерийском выстреле 

Fig. 1. The dependence of the total heat transfer coefficient on time for a small-caliber artillery shot

 

Prospects for further dissemination of solutions for AAW research

The practical application of the developed tools for assessing the intensity of thermal loading of the barrel contributes to the improvement of quantitative approaches to solving problems indirectly or directly related to the thermophysical aspects of modernization of structural schemes and algorithms for using AAW. Including the proposed model in the formalizations of a higher level of the hierarchy eliminates the need to operate with rather voluminous constructs to express the connections formed in the case of an absolutely perfect formulation of the studied problem.

The primary importance of the identified quantitative parameters, which have a major impact on the heating and cooling of the barrel, is justified by the presence of additional opportunities when working out the requirements for AAW innovations (modernizations) various automation schemes.

However, as before, there are still some questions:

– removal of a number of assumptions when calculating the gas temperature T₁, the values of which are included in the form of variables in the approximating analytical dependencies (8)–(22), which form the criterion equation (3) through expressions (6) and (7);

– impossibility of fully taking into account the dependence of the thermophysical parameters of air on the altitude of the use of AAW in the criterion equation (5) due to the theoretical complexity and significant resource intensity of dynamic mathematical models of the atmosphere.

 

Conclusion

Using the methods and techniques of the theory of thermodynamic similarity, one of the possible variants of the model has been synthesized, which makes it possible to fairly objectively identify fast processes of heat transfer during the movement of gases in the barrel bore and air near the outer wall of the barrel. It should be expected that further accumulation of knowledge of the theory of heat transfer will follow the path of a deeper study of the phenomena and a more accurate mathematical description of AAW in a thermodynamic formulation. Based on these positions, directions to improve the recommended approach are justified.

The introduction of the proposed model into the practice of designing and testing AAW will allow to obtain objective assessments of the quality of AAW operation while minimizing the plans of full-scale experiments, reducing the time and material resources for research and development.

Sobre autores

Ilya Podkopaev

Air Force Military educational and scientific center “Air Force academy named after professor N. E. Zhukovsky and Y. A. Gagarin”

Email: ilya.podkopaev.96@bk.ru

adjunct

Rússia, Voronezh

Aleksandr Podkopaev

Air Force Military educational and scientific center “Air Force academy named after professor N. E. Zhukovsky and Y. A. Gagarin”

Autor responsável pela correspondência
Email: aleksanpodkopaev@mail.ru

Cand. Sc., associate professor, professor of the department operation of aircraft weapon systems (and sighting systems)

Rússia, Voronezh

Vasily Dolzhikov

Air Force Military educational and scientific center “Air Force academy named after professor N. E. Zhukovsky and Y. A. Gagarin”

Email: Link707@mail.ru

Cand. Sc., associate professor, head of the department operation of aircraft weapon systems (and sighting systems)

Rússia, Voronezh

Bibliografia

Arquivos suplementares