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

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

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 theoretical 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 model, and on the other hand an analytical boundary

layer model using the boundary layer equations of Prandtl.

The results of both models describing the development 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 interior 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)

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 different importance occur. The most important process, the motion of the projectile inside of the accelerator tube, depends on the burning characteristic of the propellant used and thus on the generation of

Computational Methods and Experimental Measurements 453

gas temperature and pressure in the accelerator barrel, 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 inner 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 presented. One method is a full Navier-Stokes solution based on the conservation equations for mass, momentum, 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 combustion 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 behind 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-

454 Computational Methods and Experimental Measurements

ding on the special heat output of the burning propellant. In our cases the maximum gas pressure is between 400 and 500 MPa, and the maximum gas temperature 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 boundary layer and the turbulence the Navier-Stokes equations 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 differential 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.

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 necessary 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 fluxes at the wall surface (boundary) by

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 different 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 temperature 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. The calculations of ADAMS and KRIER [7] involve the unsteady, compressible, and turbulent boundary layer using the boundary layer equations combined with a

456 Computational Methods and Experimental Measurements

simple turbulence model. Again, numerical methods are required in order to solve the differential equations in Refs. 6 and 7.

The disadvantage of all these theoretical models is that they are time-consuming and require a great amount of computer resources. Therefore, it is desirable to have a simplified analytical solution for predicting the formation of the boundary layer in gun tube flows [11].

3.1 Equations For Boundary Layer Development The simplified formation of the boundary layer inside a gun tube is illustrated in the schematic of Figure 2. In the vicinity of the base of the projectile, an unsteady boundary layer is formed at the wall of the gun barrel, called "projectile boundary layer". It develops downstream as the projectile is accelerated down the tube. In addition, a wall boundary layer originates at the breech: "breech boundary layer". The entire boundary layer formation can be described, if one couples both the projectile and breech related boundary layers at equal boundary

layer thickness.

Breech Projectile

y Boundary Layer V x ___ _^__—-7—_ x

— At,

Fig 2. Schematic of the tube contours and of the boundary layer formation in the analytical model

The formation of the gun tube boundary layer is treated in two dimensions which is justified in case if the boundary layer thickness is small compared with the bore diameter. Then, Prandtl's boundary layer equations are used in two dimensions to deve- lop some relations which describe the boundary layer formation. The unsteady, time-dependent boundary layer development between breech and projectile will be calculated by using a time-step procedure with successive time intervals Ati, At2,...,Atn. During these time intervals the problem can be treated as a stationary one, wherefore the time-dependence can be withdrawn in Prandtl's equations. Then, the folio-

Computational Methods and Experimental Measurements 457 wing equation can be obtained for the wall shear stress TW, see OERTEL [8]:

5 du p6 c = d_ f pu (u - u) dy + —- [ (PgUg - pu) dy. (3) N AV J m ^ dx ^ o

The experimentally determined velocity profiles in turbulent boundary layers are given approximately by the power-law equation [5]: — \ l/n (4)

In equation (4) n is a function of the Reynolds number and ranges from 5 < n < 10. Introducing equation (4) into (3) for both the breech and the projectile boundary layer two differential equations can be obtained with the following conditions inside

the outer core flow: flow velocity increases linear- ly from breech (u = 0) to projectile (u = UP) , gas pressure p and temperature T are assumed to be constant along the tube axis for the given time intervals Ati, At2,...,Atn. From time step to time step the quantities change in time with proceeding projectile displacement XP = Xp(t) : p = p(t), T = T(t) .

Integration of the differential equations yield analytical solutions for the boundary layer thickness 6 and the heat flux qg at the tube surface, see [11]. The solutions for the projectile boundary layer are as follows in the x-coordinate system be-

ginning at the projectile base as x = XP - x.

_ , , 1^3 f L6 "In* 3 f 2 ln+ MX) - B(n) »>] [-T ) IT J

^ cp (Tr - TW ) Pr

2 2 n+ : rvf3

f IH L. l"+3 U "*' (6) [ L 6 J P x L

458 Computational Methods and Experimental Measurements

The reference frame of the breech boundary layer is located with the x-coordinate zero point at the breech. The result for that boundary layer is given as follows:

B(n) p _; P L

n+l

V*> - " »<")»<=„ <*, - v

•'' \ I'"'"

3.2 Wall Temperature Calculation Owing to small variations of the calculated heat flux qg along the axial coordinate given by equations (6) and (8), the heat flux q*(y,t) inside the tube wall is considered to be approximately only a function of depth (y-direction). Therefore, the one- dimensional heat conduction equation was applied in the region of the wall:

at p c _ 2 " \"' w w oy

The boundary conditions are: inside the tube: AT^(y,t = 0) = AT^(o>,t) = 0, (lOa)

at the surface: -^ (0,t > 0) = ^— , (lOb)

w By integration with the boundary conditions one obtains the following solutions for the temperature change ATw inside the tube wall in y-depth as a function of the heat flux qw directly at the surface at y = 0, see OERTEL [9]:

AT (y,t) = VTI p c A WWW

q..(o,-c) (t-T)

Computational Methods and Experimental Measurements 459

In coupling the heat flux qw(0,t) in equation (11) with that in equations (6), resp. (8), by assuming that the heat flux at the surface at y = 0 is equal on the gas side (qg) and on the wall side (qw) with qg(t) = q*(0,t), the temperature distribution Tw(y,t) inside the gun barrel can be calculated. The procedure described is carried out for each time

interval!' Ati, At2, . . . at all x < XP for getting the temperature distribution Tw = Tw(x,y,t) inside the gun tube at each location x and y with projectile displacement xp(t) in time t.

Additionally, the calculation of wall temperature was extended to heat transfer into a tube coated at the inner surface with a thin nickel layer of thickness d. In this case the heat conduction in depth is found as a solution of the heat conduction equation (9) for the tube wall (d < y < oo) and a second one for the surface layer (0 ^ y < d):

6T X 5T —1 = - !— — - (12)

Taking into account the heat transfer boundary conditions at the contact layer at y = d, from equations (9) and (12) an analytical solution for the temperature distribution inside of the thin surface layer and a second solution inside the wall material can be found, as given by OERTEL [8].

4. TEMPERATURE MEASUREMENT IN THE 20-MM TEST-GUN

For experimental studies of bore temperature in the medium caliber range single-shot experiments have been performed in the ISL test-gun device of 20-mm-caliber (see Figure 3) . The accelerator setup, described in detail in [10], consists mainly of the combustion chamber and the gun tube which is equip-

ped with bore-holes as measuring stations Ml, M2 , M3 , ... for instrumentation, e.g. with pressure gauges and thermocouples. Typical features of the test-gun device are: The barrel is not rifled; the projectile is starting from the forcing cone; the propellant is loaded into the combustion chamber inside of a bag; the charge is ignited electrically.

For yielding information about the wall temperature distribution along the accelerator barrel, the bore-holes are instrumented with nickel-steel thermocouples which have been developed at ISL, as given in Figure 4. They consist of a steel housing and a

460 Computational Methods and Experimental Measurements

centered nickel wire with a non-conductive layer for electric insulation. At the front side of the thermocouple a galvanic nickel layer (thickness 10 up to 100 jLtm) was coated. The tube temperature is measured in depth at position of contact and not at surface.

33 mm 347mm

Combustion M3 M4 M7 g^rrel ™ chamber

Fig. 2 Setup of 20-mm-caliber ISL test-gun

Steel tube Gasket

Nickel wire Electric insulation layer

Nickel layer 5 mm Thickness: 10 -100 urn

Fig. 3 Gun barrel nickel-steel thermocouple 5. RESULTS

In the following some computational results are presented and compared with the experimental data. All computations take account of the geometry and the interior ballistics of the 20-mm-caliber test device. The core flow boundary conditions necessary for the analytical solution are based on the history

of gas pressure, gas temperature and flow velocity computed by the numerical method. The exponent value of n for the boundary layer velocity profile was set to n = 7. The Prandtl number is Pr = 0.81.

4.1 'Cold/ propellant gas flow The A 5020 propellant, a single base propellant, is used for producing a gas flow with the relatively low flame temperature of 2700 K. Figure 5a shows the wall temperature distributions measured with thermocouples with nickel layers of d = 10, 30, 50, and 100 jLtm electroplated on the steel sensor tip. These results are found in depth of 10, 30, 50 and 100 jum. The measuring position was x = 33 mm (M3) beyond the

Computational Methods and Experimental Measurements 461 initial position of the projectile base (IPP). The numerical code as well as the analytical results are fitted to the thermocouple measurement and therefore calculated with a nickel layer of 10, 30, 50, and 100 /im on steel material. The plots in Figure 5b (numerical curves) and 5c (analytical ones) display the temperature distributions calculated in the four depths mentioned. Both the temperature rises and the maximum temperatures are in very good agreement at the surface and in depth.

The drawings in Figure 6 obtain the wall temperature in depth of 10 urn over time for the three measuring positions of x = 33 mm (M3), 347 mm (M7),

and 902 mm (M9) from IPP. The maximum wall temperatures are in good agreement between the experimental results and both theoretical calculations done with 10 /Ltm nickel layer on steel. However, for the 352 mm and 902 mm positions the temperature rise coming from the numerical code, see Figure 6b, is weaker as given from the measurements of Figure 6a. The analytical boundary layer model in Figure 6c in contrast is able to describe the strong temperature increase more correctly.

The influence of the thin nickel layer coated on

steel material can be seen in comparing the curves of Figures 5b,c with the results of Figures 7a,b which are calculated with pure steel as tube wall material. In Figures 7a and b, respectively, the temperature curves at the position x = 33 mm from IPP (M3) are presented first directly at the surface and second in depth of 10, 30, 50, and 100 /Ltm. Re- garding, e.g, in Figures 5b,c and 7a,b the maximum temperatures as they appear in depth of 10, 30, 50, and 100 urn it is apparent that there are only few differences. The two-layer nickel-steel calculations of Figure 5 give in depth of contact layer (10, 30, 50, 100 jLtm) slightly higher wall temperatures as the same calculations done with pure steel (Figure 7) .

Considering these small differences, e.g. in depth of 10 /Ltm, between nickel-coated steel tube and pure steel calculations, these differences are much smal- ler than the deviations in the thermocouple experiment with about ^10%. This outcome means that the ISL nickel-steel thermocouples can be used success- fully for temperature gauging in steel barrels.

4.2 'Hot* propellant gas type For getting a 'hot' propellant gas flow the GB- Pa 125 double-base propellant is used. Its flame temperature is very high with 3750 K. Due to this high gas temperature the tube wall temperatures are

462 Computational Methods and Experimental Measurements

TIME [ms]

Fig. 5 a) Thermocouple, b) numerical and c) analytical model: 10, 30, 50, 100 jLtm nickel-layer

Computational Methods and Experimental Measurements 463

a) 6.0 TIME [ms]

b) 6.0 TIME [ms]

C) 6.0 TIME [ms]

Fig. 6 a) Thermocouple, b) numerical and c) analytical model: position 33, 352, 902 mm off IPP

464 Computational Methods and Experimental Measurements

a) 5.0 6.0 TIME [ms]

surface

§ 3,

TIME [ms]

Fig. 7 a) Numerical and b) analytical model higher than for the A 5020 propellant. Figure 8 pre- sents a series of five thermocouple signals obtained at 33 mm from IPP (M3) at equal firing conditions. The numerical and analytical results in depth of 0, 10, 30, 50, and 100 /Ltm are given in Figure 9 for pure steel. The three approaches agree fairly well in comparing the curves in depth of 10 /Ltm.

CONCLUDING REMARKS

In the present study the boundary layer formation has been investigated theoretically and experimentally, using a numerical solution of the full Navier-Stokes equations as well as an analytical one beginning with Prandtl's boundary layer equations. Both calculations yield informations on boundary layer thickness, or heat flux into the tube wall. The wall temperature is calculated for pure steel tube material, or for a thin nickel coating.

Computational Methods and Experimental Measurements 465

5.0 TIME [ms]

Fig. 8 Measurements with 10 jim nickel-layer

surface

0 a) 4.0 5.0 6.0 TIME [ms]

surface

b) 5.0 TIME [ms]

Fig. 9 a) Numerical and b) analytical model: 0, 10, 30, 50, 100 jLim depth, steel tube wall

466 Computational Methods and Experimental Measurements

The theoretical predictions are compared to experimental temperature data measured in the ISL 20- mm-caliber test accelerator with special nickel- steel thermocouples. The measuring results and both theoretical models are in excellent agreement.

REFERENCES [1] GARLOFF, J., REISER, R., 'A Contribution to the Turbulence in Interior Ballistics Flows', AIAA Paper 89-2557, 1989

[2] HEISER, R., GARLOFF, J., 'Study on Turbulence in Interior Ballistics Flows', J. Propulsion and Power, 8, 59-65, 1992

[3] BRILEY, W. H., MCDONALD,H., 'Solution of the Multidimensional Compressible Naviei—Stokes Equa- tions by a Generalized Implicit Method', J. of Com- putational Physics, Vol. 24, 372-397, 1977

[4] HEISER, R., 'Viscous Modeling of the Interior Ballistic Cycle', Fraunhofer-Institut fur Kurzzeit- dynamik, Weil am Rhein, Report EMI 10/91, Oct.1991

[5] SCHLICHTING, H. , 'Boundary-Layer Theory', McGraw-Hill, New-York, 1960

[6] MAY, H., HEINZ, C., 'Arbeit en zur Berechnung der laminaren hompressiblen Grenzschicht in einem Rohr', 5. Symp. innenball. Leistungssteigerung von Rohrwaf- fen, BICT, Swisttal-Heimerzheim, Germany, 1979

[7] ADAMS, J., KRIER, H. , 'Unsteady Internal Boun dary Layer Analysis Applied to Gun Barrel Wall Heat Transfer', Int. J. Heat Mass Transfer, Vol. 24, No. 12, 1981

[8] OERTEL, H., 'StoBrohre', Springer Verlag, Wien-

New York (1966)

[9] OERTEL, H. , 'Messungen im HyperschallstoBrohr', p. 759-848 in 'Kurzzeitphysih', Springer Verlag, Wien-New York, 1967

[10] ZIMMERMANN, K. et al., 'Influence of normal and LOVA propellant charges on 2O-mm-gun-tube erosion', Proc. of the 12th International Symposium on Ballis- tics, San Antonio, Tx., USA, 1990

[11] SEILER, F., 'Turbulent boundary layer formation in simulated gun tube flows: theory and experiment', J. of Ball., Vol. 9, No. 2, 1987