Show that the energy wasted by this jet of air, per unit time, is where m is the mass flow through the device.

Using the propulsive efficiency defined in Prob. 9.15, discuss why the propeller is the most efficient propulsion device, the rocket engine is the least efficient, and the gas turbine jet engine is in between. Continuing the line of thought started in Prob. 9.14, the propulsive efficiency, denoted by T/p, is defined as Using the result obtained in Prob. 9.14 and recalling the definition of power available from Ch. 6, show that Prob. 9.14 Examine the control volume sketched in Fig. 9.14d. Although this control volume was used in Sec. 9.4 to obtain the thrust equation for a jet propulsion device, it can be used in general to any propulsive device that creates an increase in flow velocity through the control volume. For example, you could imagine a reciprocating engine-propeller combination inside the control volume, with air at velocity V~ coming into the control volume ahead of the propeller, and with velocity V. leaving the control volume behind the propeller. So the control volume in Fig. 9.14d is generic and can represent both propeller and jet engines. Imagine that you are standing outside in the still air and the propulsive device represented by Fig. 9.14d flies past you at velocity V ~· The inlet and exit velocities, V~ and V., shown in Fig. 9.14d are relative to the device. The exhaust velocity exiting the device relative to you is not the same as v. in Fig. 9.14d. Before the propulsive device entered your space, the air around you was still. After the device left your space, it left behind a jet of air moving in the opposite direction at a velocity different than V.. This jet of air has energy, and that energy is wasted; it performs no useful work. Show that the energy wasted by this jet of air, per unit time, is where m is the mass flow through the device.

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