The ascending trajectories performance and control to minimize the heat load for the transatmospheric aero-space planes (1990)

The paper discusses the performance and control of ascending trajectories for Transatmospheric Vehicles (TAV) using air-breathing propulsion. The work looks to minimize the heat load per unit area near the stagnation point. The vehicle is modelled as a point variabie mass with drag polar and variable thrust. The earth is assumed spherical with exponential atmosphere. The research is structured in two parts. The first part is an analytical and seeks the two controls (aerodynamic and thrust) which are necessary to transfer this TAV from one specified state(e.g. h, = 20 km, MI= 5, Q, = 5 kj/cm ,... etc.) to a second specific stale (e.g. h2= 63 km, M, = 25, Q2 = 300 kj/cm 2... etc.) while satisfying given equality constraints such as constant dynamic prcssurc, and conslant rate of climb. Certain algebraic relations among the state variables u, q, y, p, and Q and a fccdback for aerodynamic and thrust controls have been dcrivcd a closed form solutions. The heat load can be Q I 350kj/cm2 for rc 2 35 m/sec, q 5 0.2 atrn., 0 5 r, 5 1, and 0 I h, 5 1. This analytical approach was hclpful in the numcrical approach in the second pan. The second part was carried out by using extensive numcrical optimization algorithms. The conuol laws which minimize the total heat load were found as combinations of some state variables in a parametric way which give the heat load Q =300 kjl cm2 for the ~ w o controls; . s , = a , + a 2 u 2 + a 3 q , and h ,=b, + b2u2+ b3q. A numerical example was worked out for illusuation, and in general all the constraints were satisfied. a (mIscc2) acceleration component in direction of V. a , , a,, a, cocfficicn~s as an optimization design paramctcrs for the total numcrical thrust control law. where r, = T g / w o = a l + + u 2 + a l q b , , b2, b,coefficients as an optimization design para* Associate Professor of Aerospace Engineering. ** Ph. D. student of Aerospace Engineering. Member AIAA. meters for the total numerical aerodynamic control law. where A n = [ C 1 , / C ; , ] = b l + b 2 ~ 2 + b 3 q p = 0.000147 (l/m) reciprocal of b e scale height. C = 1/I = (112250. sec.) specific fuel consumption. C, = 5.73 x 108 ((wlcm2) 4 (m3/kg)( sec3 / m3)) -a constant in the heat equation, for r~ = 0.1 m (body nose radius), hw = 0.9 (wall enthalpy to total enthalpy) where C, = 1.83 x lo-' (1 h,) C2 = 1000constant for numerical purposes. C3 = 58-(m)-total length of the vehicle. C, = (p, V,2 /2 C2 C*,, )-(N/mZ)-a constant in the unit of pressure. C, = D, /q S aerodynamic drag coefficient. C*, =2 C, aerodynamic drag cocflicient at maximum aerodynamic lift to aerodynamic drag. C,, = D, /qS ram drag coefficient. C,, zero lift aerodynamic drag coefficient, where C, = C,, + Ka C2,, C, = C, + C, = (Dl / q S) total drag coefficient. C,, = L, / q S aerodynamic lift coefficient. C',, =GIaerodynamic lift coefficient at maximum aerodynamic lift to aerodynamic drag ratio. C,, = L, / q S lift coefficient due to ram drag. C,, = C,, + C,, = L, / q S total lift coefficient. dl, d, coefficient as an optimization design parameters for ram drag, where Dr / W, = [dl + d2 u2] d., coefficient as an optimi~ation design parameters ' for lift due to ram drag, where (L, / La) = d3 0, = (P v2 S Ch/2) = q S C,(Newton)Aerodynamic drag. D, = f, flh = Wo [dl + d2 u2] (Newton) ram drag. Dl = D,+ D, (Newton) total drag. E = La / D,=(C,,/C,)-aerodynamic lift to aerodynamic drag ratio. EL=Emax =((La / DJmnX =(Calr /Cads)maximum aerodynamic lift to aerodynamic drag ratio. Copyright @ 1990 by the American Institute o f Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 1 designed to spend the smallest possible time in the atmosphere which leads to small amount of heat during the ascending time. The transatmospheric vehicles(TAV) which use airbreathing propulsion are now being conceived (e.g. National Aerospace Plane (NASP) , X -30) as in Fig. 1. These vehicles are expected to accelerate for longer period of time of about 20 30 minutes in the denser part of the atmosphere (h I 70 km) to reach orbital speed. This would produce severe hating problems, particularly in the ascending phase. In [4] it was found that the heating load during entry of the TAVs is about three times the shuttle entry heating load, and during ascending phase of the TAVs the heating load is about three times the entry load (i.e. the heating lo& of a TAV during ascent is about nine times greater than the heating load of the shuttle entry).This represents a serious issue for any TAV. This paper deals with the performance and control of the ascending phase trajectories for TAVs, looking to minimize the heat load pcr unit area near the stagnation point. The research is structured into two parts. The first part is an analytical approach to scck two controls (acrodynamic and thrust) which arc ncccss,ary to transfer this TAV from one specified slate (e.g. hl, = 5, h, = 20 km, Q, = 5 kj/ cm2,.. .etc.), to a second specific state (e.g. M2 = 2 5, h2 = 63 km, Q2 = 320 kj/ crn2, etc), while satisfying given equality constrain~r such as constant dynamic pressure, and constant rate of climb. This analytical study is framed to produce the maximum generality for general TAV including high aerodynamic lift to aerodynamic drag ratio designs such as wave riders. The approach is similar to the study in [5] which was for reentry problems with only acrodynarnic lift as the trajectory controller. Here we have additional equations and two controls, to include variablc mass and thrust. The second part of this research is a trajectory optimization by numerical approach to find the suboptimal controllers (aerodynamic and thrust) to minimize the total heat load. It is similar to [6] where trajectories are optimized for aerodynamic and thrust controller while satisfying many selected equality and inequality physical and mission constraints such as limitation in spced, power, and altitude. This approach is necessarily carried out using extensive numerical computation. The results of this work are part of [7] which provides insight and specific oonstraints on thc design and operation of TAV class vehicles, and defines additional work needed in the area of trajectory sclcction including particularly high aerodynamic lift to aerodynarn~c drag ratio designs such as wave riders. The vehicle is modelled as a point variable mass with drag polar and variable thrust. The earth planet is assumed spherical and non-rotating with an exponential atmosphere. The uajectory was taken in the equatorial plane. These last assumptions could be relaxed for the detailed studies of specific missions. Fig. 2 shows the sketch of the system where the equations which describe the system are given by (1-a)-(7-a): d r/d t = V sin y rate of climb-(1-a) d p / d t = p j 3 V s i n y -(2-a) d V / d t = a = ( T , D , ) / m g s i n y -(3-a) V d y /d t = L, /m g cos y + (v2/r) cos y -(4-a) d Old t = (V/r) cos y -@a) dm/dt=-CTg/g, ,=-Tg/(I go) -(6-a) d Q/d t = C, p0.5 V3 heat rate -(7-a) 4=(1/2) p v 2 dynamic pressure-(8-a) Introducing the dimensionless variables;