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Computational Methods and Measurements of Heat Transfer to the Barrel of a Hot Gas Accelerator Under High Pressure

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Computational Methods and Measurements of Heat Transfer to the Barrel of a Hot Gas Accelerator Under High Pressure Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Computational methods and measurements of heat transfer to the barrel of a hot gas accelerator under high pressure R. Heiser", F. Seiler\ K. Zimmermann\ G. Zettler^ ^Fraunhofer-Institut fur Kurzzeitdynamik, Weil am Rhein, Germany 5 Saint-Louis, France ABSTRACT The action of the hot and compressed propellant gas flow in a gun barrel behind a projectile on heat transfer to the barrel wall has been investigated by experimental and theoretical methods. For theoreti- cal description of the in-bore flow development two interior ballistic models will be applied: On the one hand a numerical model for solving the full Navier-Stokes equations including a turbulence mo- del, and on the other hand an analytical boundary layer model using the boundary layer equations of Prandtl. The results of both models describing the deve- lopment of the compressible, turbulent boundary layer at the inner tube wall are compared with heat transfer measurements performed by use of special thermocouples in the ISL 20-mm-caliber test gun. These measurements are used to check the applicabi- lity of both in-bore flow models at realistic inter- ior flow conditions. NOMENCLATURE B factor depending on n, see [5] ci specific heat of coating (nickel) CP specific heat at constant gas pressure cw specific heat of tube material (steel) Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 452 Computational Methods and Experimental Measurements IPP initial position of the projectile base d thickness of surface layer at inner tube wall L distance from breech to projectile n factor in boundary layer velocity law p propellant gas pressure Pr Prandtl number gg heat flux from gas to tube wall qw heat flux into the tube wall r radial coordinate, zero point at tube axis Re Reynolds number t time T temperature in the propellant gas To maximum gas temperature in a cross-section Ti temperature in the coated layer Tw temperature in the tube wall Tr recovery temperature u axial flow velocity UP projectile velocity x axial coordinate, zero point at breech x axial coordinate, zero point at projectile base y radial coordinate for breech boundary layer y radial coordinate for projectile boundary layer XP projectile position 6 fluid velocity boundary layer 6* boundary layer displacement thickness S** boundary layer momentum thickness c emissivity of the wall surface Ag thermal conductivity, gas Ai thermal conductivity, coating Aw thermal conductivity, tube material p gas density pi density of coating pw density of tube material TW shear stress exerted by fluid on wall (p factor considering the gas compressibility [11] cr Stefan-Boltzmann constant Indices 2 free stream 1 coated layer w tube wall p projectile base 1. INTRODUCTION During the interior ballistic cycle in a hot gas accelerator under high pressure, processes of dif- ferent importance occur. The most important process, the motion of the projectile inside of the accelera- tor tube, depends on the burning characteristic of the propellant used and thus on the generation of Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 453 gas temperature and pressure in the accelerator bar- rel, e.g. a gun tube. Important points of view are, for instance, the life-time of such gun tubes which is strongly coupled to wear and erosion. Here the generation of heat by propellant burning and heat transport processes play an important role. The transport processes of heat conduction, convection and radiation in gas phase as well as heat transfer to tube wall should be considered. The objective of our paper is to investigate the heat transfer from the hot propellant gas to the tube wall and its heating by this process. Since we want to describe the time-dependent local heat transport processes the formation of the boundary layer has to be considered precisely. Inside this wall layer the velocity increases from zero at the barrel wall to center core velocity, and the gas temperature changes from the temperature at the in- ner surface to the higher gas temperature in the core flow of the propellant gas. This means that viscosity and heat conduction are dominant in these near wall layers causing considerable heat fluxes towards tube wall. For the theoretical description of the formation of the turbulent boundary layer two methods are pre- sented. One method is a full Navier-Stokes solution based on the conservation equations for mass, momen- tum, and energy. Turbulence is included by a one- equation model. This set of differential equations demands a complex numerical solution. The second method is an analytical one and starts from Prandtl's boundary layer equations. It allows to estimate quickly the influence of important parame- ters as well as the history of local heat flux and tube temperature versus acceleration cycle. 2. NUMERICAL SOLUTION The particular tube flow investigated is an ex- treme type of a gas flow. Initially the gas is quiescent, but it changes rapidly its state by com- bustion of a propellant (solid, liquid or gaseous) such that the gas is highly compressed and heated. After the projectile has begun to move the gas be- hind the projectile is compressed as well as expan- ded for a while until later on the expansion domina- tes. The flow is highly accelerated up to velocities of about 1000 m/s on a distance of less than 3 m in our case. The gas pressure goes up to several 100 MPa and gas temperatures up to several 1000 K depen- Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 454 Computational Methods and Experimental Measurements ding on the special heat output of the burning pro- pellant. In our cases the maximum gas pressure is between 400 and 500 MPa, and the maximum gas tempe- rature between 2700 and 3800 K. However, this high temperature acts only for a few milliseconds on the surrounding walls from metal, usually steel. There is no doubt that the flow turns turbulent causing an increased connective heat flux. The heat exchange from the hot gases to the colder wall is driven by the temperature gradient in the boundary layer and it is important to include the formation of the boundary layer in the mathematical solution. 2.1 Navier-Stokes Approach of the Tube Flow In order to include the formation of the bounda- ry layer and the turbulence the Navier-Stokes equa- tions are used to model the whole gas flow region. This kind of approach allows a feedback between the boundary layer and the center core flow as well as between the boundary layer and the tube wall. The problem is simplified by choosing an axisymmetric geometry and a constant diameter tube (Fig. 2). The governing equations for the gas flow are explained in detail in Refs. 1 and 2. There, the experimental verification of the mathematical model is also dis- cussed. The turbulence is modelled by one differen- tial equation for the turbulent kinetic energy. The combustion of the propellant is simulated by both mass and energy sources depending on the particular burning characteristics of the propellant. A typical spatial distribution of the axial flow velocity is shown in Figure 1 for UP = 1021 m/s. PRO J. DISPLACEMENTS. 100 m PROJ.VEL.=1021 rr/s r—' cs /V A/'/WX//y/A/V\ 1 1 JVllL o 1 Hr — i y 1 > S] ^####\ TIME= 7.893ms s 1 X 0.0 Fig. 1 Axial flow velocity in a 20-mm-tube at projectile exit. Propellant A 5020. Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 455 The numerical method is based on a linearized ADI scheme [3] applied to a moving coordinate system caused by the moving projectile. The implicit finite difference method is very well suited since a satis- factory resolution of the boundary layer makes ne- cessary very small grids in the wall proximity. To keep the amount of grid points as small as possib- le, a distribution of non-equidistant grids is used in radial direction with most narrow grids at the tube wall and increasing grid size in direction to the tube axis. In axial direction equidistant grids are sufficient. 2.2 Heat Transfer The heat exchange between the hot gases and the cold wall is calculated by the balance of heat flux- es at the wall surface (boundary) by 9T assuming T = Tw at the boundary. The balance of heat fluxes includes a term for radiated heat. In the wall the heat conduction is considered according to 9T W O, , ,. 9T VK , J. d , -. dTW. / ,) \ P. °w at- = H <*w aF> + r a? ^ ^w aF> ^) The thermal properties Aw and cw depend on r and x and also on Tw. In case the tube consists of layers or segments of different kinds of material, pw depends on r and x. Very often the inner surfaces are protected by a thin layer of a material dif- ferent from the main tube material. For completion the boundary conditions a simple engineering condition is chosen at the outer surface whereas at the projectile position ambient tempera- ture and at the rear end an adiabatic condition seems to be adequate [4]. 3. ANALYTICAL MODEL FOR BOUNDARY LAYER PREDICTION If only the boundary layer formation is to be taken into account, then simplified conservation equations [5] can be applied. Such calculations have been carried out by MAY and Heinz [6] for the case of a compressible and laminar tube boundary layer.
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