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Chapter05

# Chapter05 - Conservation of Mass and Momentum Momentum...

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Conservation of Mass and Conservation of Mass and Momentum Momentum z Using the Reynolds Transport Theorem, it is a simple matter to derive mass- and momentum-conservation laws for a control volume, i.e., the integral forms Global view Indirect computation of forces z Focusing on a differential-sized control volume, we can deduce the differential form of the conservation laws, which hold at every point in the flow Detailed view Direct computation of forces and all flow properties z From the differential forms we can derive Bernoulli’s Equation – mechanical energy conservation

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Conservation Principles Conservation Principles z The basis of our conservation laws is as follows, where B is an extensive variable and β is the corresponding intensive variable z We will concentrate on mass and momentum first, and do energy later (Chapter 7)
Mass Conservation Mass Conservation - - 1 1 z We consider a general control volume whose bounding surface has velocity u cv z Definition of a system Mass, M , is constant By definition, Hence, B = M and β = 1 z dM/dt = 0 …the Reynolds Transport Theorem yields z In words, the sum of the instantaneous rate of change of mass in the control volume and the net flux of mass out of the control volume is zero

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Mass Conservation Mass Conservation - - 2 2 z Recall that for a differential-sized control volume… z Again, since dB/dt = 0 and d β /dt = 0 , the Reynolds Transport Theorem tells us that z This is called the continuity equation…it holds at every point in the flow – CV asymptotes to a point z Using the chain rule… “Conservation” form “Primitive- variable” form Æ
Mass Conservation Mass Conservation - - 3 3 z We can use our results to further simplify the Reynolds Transport Theorem…as noted above, for a “point” we derived the following z Therefore,the Reynolds Transport Theorem for a “point” simplifies to Zero according to mass conservation

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