# lec4 - Chapter 4 Control Volume Analysis Using Energy...

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1 Chapter 4: Control Volume Analysis Using Energy Conservation of Mass and Conservation of Energy The Reynold Transport Theorem where β is the specific property corresponding to B. Then n V V n r r = Where is the velocity normal to the control surface and dB CV /dt is the rate of change of B within the control volume: Let B be an extensive property of a system: Ω βρ = dV t B ) ( sys βρ + = CS n CV dA V dt dB dt dB sys ( ) dV t dt dB CV CV βρ = n r CV CS Ω CS: Control Surface CV: Control Volume Ω =CV at time t

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2 RTT Proof Sketch CV sys ) ( ) ( sys t B t B CV = At time t: B t t B t t B CV Δ + Δ + = Δ + ) ( ) ( sys At time t+ Δ t: () tdA V B d n Δ βρ = Δ Flux: n r 0 < n V 0 > n V - + t dA V B CS n Δ βρ = Δ d( Δ B) RTT Proof Sketch CV sys ) ( ) ( sys t B t B CV = At time t: B t t B t t B CV Δ + Δ + = Δ + ) ( ) ( sys At time t+ Δ t: Flux: n r t B t t B t t B t t B t t B CV sys Δ Δ + Δ Δ + = Δ Δ + ) ( ) ( ) ( ) ( CV sys t dA V B CS n Δ βρ = Δ Δ t 0 βρ + = CS n CV dA V dt dB dt dB sys
3 Conservation of Mass • A special case of RTT is when β =1 and B=m, and dm/dt=0. • The RTT becomes • Let the CS area such that V n is negative across A i (inlet) and positive across A e (outlet) then ρ = CS n CV dA V dt dm + = e i A A A e e i i CV A V A V dt dm ρ ρ = where V i =-V n >0 across A i and V e =V i >0 across A e . Conservation of Mass for a Control Volume time of time of flow time of flow mass contained within of mass across of mass across the control volume inlet exit rate of change rate rate in out at time t i at time t e at time t ⎡⎤ ⎢⎥ =− ⎣⎦ cv ie dm mm dt where: n A mV d A ρ =

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4 Other forms of the Conservation of Mass ie mm = Steady-State Flow: Integral Form: ( ) ( ) nn VA A d dV V dA V dA dt ρρ ρ =− ∑∑ ∫∫ One-Dimensional Flow: AV mA V v == cv i i e e dm AV AV dt v v Example Problems • Problem 4.4 • Problem 4.7 • Problem 4.13
5 Problem 4.4 Problem 4.7

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lec4 - Chapter 4 Control Volume Analysis Using Energy...

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