p06112 - 742 CHAPTER 6 CONTROL-VOLUME METHOD 6.112 Chapter...

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742 CHAPTER 6. CONTROL-VOLUME METHOD 6.112 Chapter 6, Problem 112 Figure 6.112: Small boat powered by an air jet. 6.112(a): Let the control volume be coincident with the surface of the boat. It remains fixed with the boat, and is thus accelerating. Also, let the positive x direction be to the right. Conservation of mass tells us d dt 888 V ρ dV + 8 s 8 S ρ u rel · n dS = 0 By definition, the mass of the boat, including the air in the compressor, is M , and the only place fluid crosses the control-volume surface is the air jet. Thus, M 888 V ρ dV and 8 s 8 S ρ u rel · n dS = π 4 ρ e u e d 2 Combining these two equations, we have dM dt + π 4 ρ e u e d 2 = 0 Turning to momentum, the conservation principle tells us that d dt 888 V ρ u dV + 8 s 8 S ρ u p u rel · n Q dS = i · 8 s 8 S ( p p a ) n dS F D where F D is the hydrodynamic drag on the boat, and the absolute velocity is u = U ( t ) + u rel where u rel is the velocity relative to the boat. Hence, the unsteady term is d dt 888 V ρ u dV = d dt 888 V ρ U dV + d dt 888 V ρ u rel dV
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6.112.
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