# Rates of momentum flow into and out of the control

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rates of momentum flow into and out of the control volume are a V m and 5 V m respectively. The net force exerted by the exit pressure is 5 5 ) ( A p p a - , where 5 A is the nozzle exit area. Thus, applying Newton’s Second Law to the control volume, we can relate the force exerted by the engine on the gases flowing through, n F , and the net exit pressure force to the rate of increase of flow momentum produced by the engine: Figure 5

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Gross thrust, g F ; 5 V m F g = Net Thrust, n F a a g n V m F F - = 5 5 5 ) ( } ) 1 {( A p p V V f m F a a a n - + - + = (1) Fuel-air-ratio f ; a f m m f = Total mass flow; a f m m m + = Here, n F is the engine force acting on the gas through flow. The reaction to this force is the thrust on the engine acting in the direction of flight. Thus the magnitude of the thrust is given by Equation (1). It is the sum of all the pressure force components acting on the inside the engine in the direction of flight. The exit area, 5 A , is related to the mass flow rate by 5 5 5 V A m ρ = (2) where the density at station 5 is obtained from the perfect gas law using 5 p and 5 T . If 5 A is known, the mass flow rate through the engine may be determined from Equation (2) and the thrust from Equation (1). Another type of nozzle used in high-performance engines and in rocket nozzles is a convergent-divergent nozzl e, one in which the flow area first contracts and then increases. It differs from the convergent nozzle in that it can have supersonic flow at the
exit. For such a fully expanded, convergent-divergent nozzle operating at its design conditio n, a p p = 5 , and the engine thrust from Equation (1) reduces to; } ) 1 {( 5 a a n V V f m F - + = (3) Jet Engine Performance It is seen that engine thrust is proportional to the mass flow rate through the engine and to the excess of the jet velocity over the flight velocity. The specific thrust of an engine is defined as the ratio of the engine thrust to its mass flow rate. From Equation (2) the specific thrust is m A p p V V m F F a a n s / ) ( ) ( 5 5 5 - + - = = If a p p = 5 ) ( 5 a n s V V m F F - = = Because the engine mass flow rate is proportional to its exit area, as seen in Equation (2), m A / 5 depends only on design nozzle exit conditions. As a consequence, s F is independent of mass flow rate and depends only on flight velocity and altitude. Assigning an engine design thrust then determines the required engine-mass flow rate and nozzle exit area and thus the engine diameter. Thus the specific thrus t, s F , is an important engine design parameter for scaling engine size with required thrust at given flight conditions. Another important engine design parameter is the thrust specific fuel consumption TSFC, the ratio of the mass rate of fuel consumption to the engine thrust n f F m TSFC = [kg/kN-s]

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Low values of TSFC, of course, are favorable. The distance an aircraft can fly without refueling, called its rang e, is inversely proportional to the TSFC of its engines.
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