## M0 mf mp210

When applied to a rocket propulsion system, the mass ratio IVR and propellant fraction f are different from those that apply to a vehicle as described above. Here the initial or loaded mass m0 consists of the inert propulsion mass (the hardware necessary to burn and store the propellant) and the effective propellant mass. It would exclude masses of nonpropulsive components, such as payload or guidance devices. For example, in a liquid propellant rocket engine the final or inert propulsion mass mf would include the propellant feed tanks, the pressurization system (with turbopump and/or gas pressure system), one or more thrust chambers, various piping, fittings and valves, an engine mount or engine structure, filters and some sensors. The residual or unusable remaining propellant is usually considered to be part of the final inert mass mf, as it will be in this book. However, some rocket propulsion manufacturers and some literature assign residuals to be part of the propellant mass mp. When applied to a rocket propulsion system, the value of the propellant mass fraction f indicates the quality of the design; a value of, say, 0.91 means that only 9% of the mass is inert rocket hardware and this small fraction contains, feeds, and burns a substantially larger mass of propellant. A high value of f is desirable.

The impulse-to-weight ratio of a complete propulsion system is defined as the total impulse /, divided by the initial or propellant-loaded vehicle weight w0. A high value indicates an efficient design. Under our assumptions of constant thrust and negligible start and stop transients, it can be expressed as h = I, w0 (mf + mp)go Is mf/mp + 1

The thrust to weight ratio F/w0 expresses the acceleration (in multiples of the earth's surface acceleration of gravity) that the engine is capable of giving to its own loaded propulsion system mass. For constant thrust the maximum value of the thrust to weight ratio, or maximum acceleration, occurs just before termination or burnout because the vehicle mass has been diminished by the

mass of useful propellant. Values of F/w are given in Table 2-1. The thrust to weight ratio is useful to compare different types of rocket systems.

Example 2-1. A rocket projectile has the following characteristics:

Initial mass 200 kg

Mass after rocket operation 130 kg

Payload, nonpropulsive structure, etc. 110 kg

Rocket operating duration 3.0 sec

### Average specific impulse of propellant 240 sec

Determine the vehicle's mass ratio, propellant mass fraction, propellant flow rate, thrust, thrust-to-weight ratio, acceleration of vehicle, effective exhaust velocity, total impulse, and the impulse-to-weight ratio.

SOLUTION. Mass ratio of vehicle (Eq. 2-8) \R = mf/m0 = 130/200 = 0.65; mass ratio of rocket system 1VR = mf/m0 = (130 - 110)/(200 - 110) = 0.222. Note that the empty and initial masses of the propulsion system are 20 and 90 kg, respectively. The propellant mass fraction (Eq. 2-9) is

The propellant mass is 200 — 130 = 70 kg. The propellant mass flow rate is m = 70/3 = 23.3 kg/sec, The thrust (Eq. 2-5) is

The thrust-to-weight ratio of the vehicle is initial value F/w0 = 54,857/(200 x 9.81) = 28 final value 54,857/(130 x 9.81) = 43

The maximum acceleration of the vehicle is 43 x 9.81 = 421 m/sec2. The effective exhaust velocity (Eq. 2-6) is c = Isg0 = 240 x 9.81 = 2354 m/ sec

This result can also be obtained by multiplying the thrust by the duration. The impulse-to-weight ratio of the propulsion system (Eq. 2-11) is 