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Aviation History
1959
1959 - 2905.PDF
504 FLIGHT, 6 November 1959 Above, workers in General Electric's Missile and Space Vehicle Department are inspecting a Mk 2 heat-sink nosecone. A 1,115 Ib copper slab, it has a nose radius of 14.94in, and for the Atlas ICBM has a nickel-plate surface polished to a finish of 72 micro-inches. It is now being replaced by the lighter and faster- flying Mk 3 cone (left) which is shown in the drawing of Atlas on page 508 BASIC BALLISTICS . . . A method of preventing such tumbling is to spin the vehicle about its longitudinal axis so that the resultant gyroscopic forces maintain the desired angle. In a well-stabilized trajectory the angle of attack of the vehicle does not change outside the atmo- sphere. It follows the elliptic trajectory, and reaches a maximum altitude of the order of 2 X 10*ft (380 miles) for a typical long-range ballistic missile. Re-entry. The density of the atmosphere fades out relatively gradually, so that it is not possible to define the exact instant at which a vehicle re-enters. Experience seems to suggest that for our purpose one can neglect the effect of that part of the atmo- sphere above roughly 250,000ft (—50 miles). When the nose- cone reaches this height on the descent it is travelling very slightly faster than at burnout, since it is lower than burnout height. The vital question "What happens to a re-entering vehicle?" is best answered by studying the work of Allen and Eggers, and in particular Ref. 4. The problem will be broken down into two parts. We first study the motion involved and then apply the resulting information to a study of the aerodynamic heating of the re-entry body. From Ref. 4, the equations for velocity and acceleration are, neglecting gravity -dV ~dt~" V = VE e 2pm sin CDPO^FE' -to _ 2mg e e -tofi m sin (9) (10) where VB is the entry velocity; ft is 1/22,000ft-1 and is the con- stant in the expression for density (i.e., e=0.0034 e-vaowjj y is the height; m is the mass of the body; and SE is the re-entry angle. The other symbols have their usual aeronautical meanings. One of the basic assumptions of (9) and (10) is that the drag coefficient is constant. This may at first seem strange, but for the very high speeds involved it is an entirely adequate assump- tion for a comparative analysis of this type. The neglect of gravity is justified since deceleration due to aerodynamic drag is rela- tively large. This means that the re-entry trajectory is virtually a straight line, and the re-entry angle is maintained throughout. A significant point about (10) is that the deceleration is pro- portional to (entry velocity)2, as is the total heat input to the body. The reasons for the latter proportionality are discussed after (15). Referring back to Fig. 2 it is seen that burnout and hence entry velocities go up with range. Thus the longer the range of the vehicle the more severe are the re-entry problems; a nosecone designed for a range of 1,000 miles cannot be attached to an ICBM and expected to survive re-entry. Investigation of the maximum deceleration reached reveals some surprising facts. It is given by = 026x2ge dent of all the physical and dimensional characteristics of the body. Thisdecelerationisreachedataheightof (22,000 fog22'0000""0 A) m sin »E / feet, and the velocity at this time is 61 per cent of the re-entry velocity. Clearly negative heights have no significance here. These facts enable non-dimensional velocity and acceleration charts to be produced in Ref. 4 and they are shown here as Fig. 5. In addition, the ordinate and abscissa show the dimensional values pertaining to our vehicle (i.e., VE of 22,000ft/sec and % of 23 deg). The maximum deceleration is 49g. Assuming a drag coefficient of 2.0 and a frontal area of 12.5ft2 gives the height of maximum deceleration as 103,000ft. In practice, calculations would be com- plicated by the fact that the body would be oscillating slightly. Before discussing heating during re-entry it is worthwhile to introduce a few ideas about aerodynamic heating in general. It is well known that the air molecules just outside the boundary layer are moving at speeds of the same order of magnitude as the free- stream velocity, while those in contact with the body are at rest. (This neglects "slip" flow, which can occur in very rarefied atmo- spheres at high speeds.) As usual in this type of problem the velocities are measured relative to the re-entry body. At hyper- sonic speeds the viscous or shearing stresses in the boundary layer must be very large to produce such a velocity-gradient across this thin layer. This results in a considerable portion of the kinetic energy of the air being transformed into thermal energy, which heats up the boundary layer. The air temperature at a stagnation point is, in the absence of heat transfer L -T- J (12) Equation (11) shows that the maximum deceleration is indepen- where T is the absolute free-stream ambient temperature, Tw is the absolute wall temperature (so called because it is the tem- perature the wall will reach when equilibrium—no heat-transfer between boundary layer and structure—is attained), M is the Mach number, T the ratio of specific heats (roughly 1.4) and r the recovery factor (which can be explained as follows). A certain percentage of the heat is conducted into the free stream, and thus not all of it is available to heat the boundary layer. Hence the temperature in the boundary layer does not reach the full stagna- tion value. The factor r is usually somewhere between 0.75 and 0.90, depending on the boundary-layer characteristics. Equation (12) is plotted in Fig. 6. Heac transferred per unit area per unit time to the re-entry body is given by dQ h ' Tx= jOTw-Ts) (13) where Q is the heat transferred per unit area (ft-lb/ft2), J is the mechanical equivalent of heat (ft-lb/BTU), h is the heat-transfer coefficient (ft-lb/ft2 sec ° R), and Ts is the skin temperature (° R). When Tw— T$ we have no heat transfer and thusTwis also known as the equilibrium temperature. From (12) and (13) it is possible to compute the heat transfer, once Mach number, ambient temperature and heat-transfer coefficient are known. However the computation of h, which is a function of position on the body, is a problem of staggering com- plexity and only the barest elements will be mentioned. It should a'so be noted at this stage that (12) is not correct if radiant heat
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