This chapter deals with the performance of rocket-propelled vehicles such as missiles, spacecraft, space launch vehicles, or projectiles. It is intended to give the reader an introduction to the subject from a rocket propulsion point of view. Rocket propulsion systems provide forces to a flight vehicle and cause it to accelerate (or decelerate), overcome drag forces, or change flight direction. They are usually applied to several different flight regimes: (1) flight within the atmosphere (air-to-surface missiles or sounding rockets); (2) near-space environment (earth satellites); (3) lunar and planetary flights; and (4) sun escape; each is discussed further. References 4-1 to 4-4 give background on some of these regimes. The appendices give conversion factors, atmosphere properties, and a summary of key equations. The chapters begins with analysis of simplified idealized flight trajectories, then treats more complex flight path conditions, and discusses various flying vehicles.

4.1. GRAVITY-FREE, DRAG-FREE SPACE FLIGHT

This simple rocket flight analysis applies to an outer space environment, where there is no air (thus no drag) and essentially no significant gravitational attraction. The flight direction is the same as the thrust direction (along the axis of the nozzle), namely, a one-dimensional, straight-line acceleration path; the propellant mass flow m, and thus the thrust F, remain constant for the pro-pellant burning duration tp. For a constant propellant flow the flow rate is mp/tp, where mp is the total usable propellant mass. From Newton's second law and for an instantaneous vehicle mass m and a vehicle velocity u.

For a rocket where the propellant flow rate is constant the instantaneous mass of the vehicle m can be expressed as a function of the initial mass of the full vehicle m0, mp, tp, and the instantaneous time t.

1p J

Equation 4-3 expresses the vehicle mass in a form useful for trajectory calculations. The vehicle mass ratio 1VR and the propellant mass fraction £ have been defined by Eqs. 2-7 and 2-8. They are related by

A definition of the various masses is shown in Fig. 4-1. The initial mass at takeoff w0 equals the sum of the useful propellant mass mp plus the empty or final vehicle mass mf, rrif in turn equals the sum of the inert masses of the engine system (such as nozzles, tanks, cases, or unused, residual propellant), plus the guidance, control, electronics, and related equipment, and the pay-load.

For constant propellant flow m and a finite propellant burning time the total propellant mass mp is mtp and the instantaneous vehicle mass m — mQ — mt. Equation 4—1 can be written as du — (F/m)dt - (ctn/m) dt

_ {cm) dt _ c(mp/tp) dt _ c^/tp ™o - mpt/tp m0( 1 - mpt/m0tp) 1 - ^t/tp

Integration leads to the maximum vehicle velocity at propellant burnout up that can be attained in a gravity-free vacuum. When uQ ^ 0 it is often called the velocity increment Am.

Au — —c ln(l - £) + u0 = c\n(m0/mf) + m0 (4—5)

If the initial velocity m0 is assumed to be zero, then up = Am = —cln(l — f) = —cln[m0/(m0 - mp)]

This is the maximum velocity increment Am that can be obtained in a gravity-free vacuum with constant propellant flow, starting from rest with m0 = 0. The effect of variations in c, /,, and £ on the flight velocity increment are shown in Fig. 4-2. An alternate way to write Eq. 4-6 uses e, the base of the natural logarithm.

The concept of the maximum attainable flight velocity increment Am in a gravity-free vacuum is useful in understanding the influence of the basic parameters. It is used in comparing one propulsion system or vehicle with another, one flight mission with another, or one proposed upgrade with another possible design improvement.

From Eq. 4-6 it can be seen that propellant mass fraction has a logarithmic effect on the vehicle velocity. By increasing this ratio from 0.80 to 0.90, the interplanetary maximum vehicle velocity in gravitationless vacuum is increased by 43%. A mass fraction of 0.80 would indicate that only 20% of the total vehicle mass is available for structure, skin, payload, propulsion hardware, radios, guidance system, aerodynamic lifting surfaces, and so on; the remaining 80% is useful propellant. It requires careful design to exceed 0.85; mass fraction ratios approaching 0.95 appear to be the probable practical limit for single-stage vehicles and currently known materials. When the mass fraction is 0.90, then Ml = 0.1 and 1/1VR = 10.0. This marked influence of mass fraction or mass ratio on the velocity at power cutoff, and therefore also the range,

125,000

Specific impulse, sec 200 400

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