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04%20Basic%20Laws%20EGM%206812%20F11

# 04%20Basic%20Laws%20EGM%206812%20F11 - EGM 6812 F11...

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EGM 6812, F11, University of Florida, M. Sheplak 1/24 Section 4, Basic Laws 4 Basic Laws 4.1 Review of Reynold’s Transport Theorem (RTT) Adopted from and figures from Potter and Foss Chapter 2 4.1.1 Non-Deformable RTT Starting with the basic equation for the relation between an arbitrary extensive property, N, and its corresponding intensi ve property, η: N d  From here we take the time rate of change of the system, N sys, as: 0 ( ) ( ) lim sys sys sys sys t DN N t t N t D d Dt Dt t        First we apply this formulation to a fixed, non-deformable control volume, as depicted in the following figure: Note: the shape of the control volume is intentionally arbitrary to indicate that it can be formed as the application requires. Proper selection of control volume is an essential component to solving such problems.

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EGM 6812, F11, University of Florida, M. Sheplak 2/24 Section 4, Basic Laws At time, t , both the system and control volume share the same space. As we progress in time, to t+∆t , the control volume occupies its original space, but the system has moved to a new position. Using the figure notation the terms from the previous equation are recast as: 3 2 ( ) ( ) ( ) sys N t t N t t N t t       2 1 ( ) ( ) ( ) sys N t N t N t Thus, we rewrite the equations as: 3 2 2 1 0 ( ) ( ) ( ) ( ) lim sys t DN N t t N t t N t N t Dt t       2 1 2 1 3 1 0 ( ) ( ) ( ) ( ) ( ) ( ) lim t N t t N t t N t N t N t t N t t t           Where in the second equation 1 ( ) N t t   has been added and subtracted to achieve the desired form. Once more referring to the figure above, the equation is manipulated to become: . . . . 3 1 0 0 ( ) ( ) ( ) ( ) lim lim sys c v c v t t DN N t t N t N t t N t t Dt t t           By definition, the first term on the right hand side is: . . . . . . 0 ( ) ( ) lim c v c v c v t N t t N t d d t dt      The second term is analyzed by realizing that 1 ( ) N t t   and 3 ( ) N t t   are the total quantity of the extensive property contained in their respective regions at time, t+∆t. Thus: 3 1 3 1 3 1 . . ˆ ˆ ( ) ( ) ( ) ˆ ˆ A A A A c s N t t N t t V ndA t V n dA t V ndA t V ndA t             This leads us to a form of the system to control volume transformation of . . . . ˆ sys c v c s DN d d V ndA Dt dt    , But the control volume is not deforming, so the   t    . Therefore, the derivative with respect to time has been brought into the integral in accordance with Leibniz' rule, as the control volume does not change with respect to time. This equation represents the Reynold's Transport Theorem.
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