AVAILABLE THRUST AND POWER

PRINCIPLES OF PROPULSION

All powerplants have in common certain general principles. Regardless of the type of propulsion device, the development of thrust is related by Newton’s laws of motion.

F~ma

or

F_ d(mV) dt

where

jF=force or thrust, lbs.

m=mass, slugs

a=acceleration, ft, per sec.2

^_derivative with respect to time, e. g., dt rate of change with time

mV= momentum, lb.-sec., product of mass and velocity

The force of thrust results from the accelera­tion provided the mass of working fluid. The magnitude of thrust is accounted for by the rate of change of momentum produced by the powerplant, A rocket powerplant creates thrust by creating a very large change in veloc­ity of a relatively small mass of propellants. A propeller produces thrust by creating a com­paratively small change in velocity of a rela­tively large mass of air.

The development of thrust by a turbojet or ramjet powerplant is illustrated by figure 2.5- Air approaches at a velocity, V, depending on the flight speed and the powerplant operates on a certain mass flow of air, Q, which passes through the engine. Within the powerplant the air is compressed, energy is added by the burning of fuel, and the mass flow is expelled from the nozzle finally reaching a velocity, W The momentum change accomplished bv this action produces the thrust,

Ta = Q(V2-V0

where

Ta = thrust, lbs.

X) = mass flow, slugs per sec.

Vі= inlet (or flight) velocity, ft, per sec.

V2— jet velocity, ft. per sec.

The typical ramjet or turbojet powerplant de­rives its thrust by working with a mass flow relatively smaller than that of a propeller but a relatively greater change of velocity. From the previous equation it should be appreciated that the jet thrust varies directly with the mass flow Q, and velocity change, V2—Vx. This fact is useful in accounting for many of the performance characteristics of the jet power – plant.

In the process of creating thrust by mo­mentum change of the airstream, a relative velocity, Vs—Vi, is imparted to the airstream. Thus, some of the available energy is essen­tially wasted by this addition of kinetic energy to the airstream. The change of kinetic energy per time can account for the power wasted in the airstream.

Pw=KE/t

-§ (v,-v, y

Ta = Q (V2-V,) Pq*T0V, Pw=Q/2(v2-V|) 2

Of course, the development of thrust with some finite mass flow will require some finite velocity change and there will be the inevita­ble waste of power in the airstream. In order to achieve high efficiency of propulsion, the thrust should be developed with a minimum of wasted power.

The propulsion efficiency of the jet power – plant can be evaluated by comparing the propulsive output power with the input power. Since the input power is the sum of the output power and wasted power, an expression for propulsion efficiency can be derived.

Pa

Vp V2+Vi

where

r/p = propulsion efficiency r) = “ eta’ ’

Pa = propulsive power available = TaV,

Pw = power wasted

The resulting expression for propulsion effi­ciency, vP, shows a dependency on the flight velocity, Vi, and the jet velocity, V2. When the flight velocity is zero, the propulsion efficiency is zero since all power generated is wasted in the slipstream and the propulsive power is zero. The propulsion efficiency would be 1.00 (or 100 percent) only when the flight velocity, V1} equals the jet velocity, V2. Actually, it would not be possible to produce thrust under such conditions with a finite mass flow. While 100 percent efficiency of propul­sion can not be attained practically, some insight is furnished to the means of creating high values of propulsion efficiency. To ob­tain high propulsion efficiency it is necessary to produce the required thrust with the highest possible mass flow and lowest possible velocity change.

The graph of figure 2.5 shows the variation of propulsion efficiency, ifp, with the ratio of flight speed to jet velocity, VijV2. To achieve a propulsion efficiency of 0.85 requires that the flight velocity be approximately 75 percent of the slipstream speed relative to the airplane. Such a propulsive efficiency could be typical of a propeller powered airplane which derives its thrust by the propeller handling a large mass flow of air. The typical turbojet power – plant cannot achieve such high propulsive efficiency because the thrust is derived with a relatively smaller mass flow and larger veloc­ity change. For example, if the jet velocity is 1,200 ft. per sec. at a flight velocity of 600 ft. per sec., the propulsion efficiency is 0.67. The ducted fan, bypass jet, and turboprop arc vari­ations which improve the propulsive efficiency of a type of powerplant which has very high power capability.

When the conditions of range, endurance, or economy of operation are predominant, high propulsion efficiency is necessary. Thus, the propeller powered airplane with its inherent high propulsive efficiency will always find ap­plication. The requirements of very high speed and high altitude demand very high propulsive power from relatively small power – plants. When there are practical limits to the increase of mass flow, high output is obtained by large velocity changes and low propulsive efficiency is an inevitable consequence.