p0648 - 594 CHAPTER 6 CONTROL-VOLUME METHOD 6.48 Chapter 6...

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594 CHAPTER 6. CONTROL-VOLUME METHOD 6.48 Chapter 6, Problem 48 There are two unknown quantities in this problem, which are the velocity U 1 and the pressure p 4 . So, in addition to the mass-conservation principle, we must appeal to the momentum-conservation equation to establish the pair of equations needed to solve. It will suffice to use the x component of the momentum equation. The optimum control volume lies entirely outside of the pipe system and has its bounding surface normal to all inlets and outlets. This ensures that the pressure differs from atmospheric pressure only at the inlets and outlets, and that the inlet/outlet velocity vectors are all parallel to the outer unit normals. Figure 6.48 shows the control volume as a dashed contour. y x U 1 ,p 1 1 25 U, p 4 U, p 2 2 U, p 3 30 o n 1 n 2 n 3 n 4 5 d 2 d d ..................................................................................... . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................................................................................................ .................................................................... . . . . . . . . . . . . ............. . . . . . . . . . . . . ................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. . . . . . . . . . ................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 6.48: Pipe with multiple branches and control volume. Mass Conservation: For steady, incompressible flow through the stationary control volume shown, the mass-conservation principle simplifies to 8 s 8 S ρ u · n dS =0
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p0648 - 594 CHAPTER 6 CONTROL-VOLUME METHOD 6.48 Chapter 6...

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