레이놀즈 수송정리 Reynolds Transport Theorem (RTT)

 

 

레이놀즈 수송정리 Reynolds Transport Theorem (RTT)

[1][2]

 

B를 질량, 에너지, 모멘텀과 같은 종량적 상태량 extensive property라고 하고

b를 b=B/m 으로 B의 강성적 상태량 intensive property라고 하자.

 

 

fixed control volume에서 시간 t에서 t+Δt에 따른 control volume이

아래 그림과 같이 변한다고 가정

시간 t에서 CV를 system이라고 하면

\[B_{sys}(t)=B_{CV}(t)\]

 

시간 t+Δt에서CV - I + II을 system이라고 하면

\[B_{sys}(t+\Delta t)=B_{CV}(t+\Delta t)-B_{I}(t+\Delta t)+B_{II}(t+ \Delta t)\]

 

시간 t+Δt와 t의 차이인 Δt 에 대한

\[B_{sys}(t+\Delta t)-B_{sys}(t)\]

의 변화량은

 

\[\frac{B_{sys}(t+\Delta t)-B_{sys}(t)}{ (t+\Delta t)-t}\]

Δt->0 일 때 위 식의 limit 값은

\[\lim_{\Delta t\rightarrow 0} \frac{B_{sys}(t+\Delta t)-B_{sys}(t)}{ (t+\Delta t)-t}\]

\[=\lim_{\Delta t\rightarrow 0} \frac{\left \{ B_{CV}(t+\Delta t)-B_{I}(t+\Delta t)+B_{II}(t+\Delta t) \right \}-B_{CV}(t)}{ (t+\Delta t)-t}\]

\[=\lim_{\Delta t\rightarrow 0} \frac{\left \{ B_{CV}(t+\Delta t)-B_{CV}(t)\right \}+B_{II}(t+\Delta t)-B_{I}(t+\Delta t) }{\Delta t}\]

 

\[\lim_{\Delta t\rightarrow 0} \frac{B_{sys}(t+\Delta t)-B_{sys}(t)}{ (t+\Delta t)-t}=\frac{dB_{sys}}{dt}\]

 

\[B_{CV}=\int_{CV}^{} \rho b dV\]

\[\lim_{\Delta t\rightarrow 0} \frac{ B_{CV}(t+\Delta t)-B_{CV}(t)}{ \Delta t}=\frac{dB_{CV}}{dt}=\frac{d}{dt} \int_{CV}^{} \rho b dV \]

 

 

\[B=bm\]

\[dB=bdm\]

 

[2]

 

\[\theta < 90^{\circ} \left ( cos\theta >0 \right )\]

 

\[dm_{out}=\rho dV\]

 

\[dV=(dl_{n})(dA)\]

\[dl_{n}=(dl)(cos\theta)\]

\[dl=(\widetilde{V})(\Delta t)\]

\[dV=(\widetilde{V})(\Delta t)(cos\theta)(dA)\]

 

\[dB_{out}=bdm_{out}\]

\[dB_{out}=b\rho(\widetilde{V})(\Delta t)(cos\theta)(dA)\]

 

\[\int_{CS_{out}}^{}dB_{out}=B_{II}(t+\Delta t)\]

\[\int_{CS_{out}}^{}dB_{out}=\int_{CS_{out}}^{}b\rho(\widetilde{V})(\Delta t)(cos\theta)(dA)\]

 

\[\lim_{\Delta t\rightarrow 0} \frac{B_{II}(t+\Delta t) }{\Delta t}\]

\[\equiv \dot{B}_{out}\]

 

\[\lim_{\Delta t\rightarrow 0} \frac{B_{II}(t+\Delta t) }{\Delta t}\]

\[=\lim_{\Delta t\rightarrow 0} \frac{1}{\Delta t}\left (\int_{CS_{out}}^{}b\rho(\widetilde{V})(\Delta t)(cos\theta)(dA) \right )\]

\[=\int_{CS_{out}}^{}b\rho(\widetilde{V})(cos\theta)(dA) \]

 

\[\widetilde{V}(cos \theta)=\vec{\widetilde{V}}\cdot \vec{n}\]

 

\[\dot{B}_{out}=\lim_{\Delta t\rightarrow 0} \frac{B_{II}(t+\Delta t) }{\Delta t}\]

\[=\int_{CS_{out}}^{}b\rho\vec{\widetilde{V}}\cdot \vec{n}(dA) \]

 

 

[2]

 

\[\theta > 90^{\circ} \left ( cos\theta <0 \right )\]

 

\[dm_{in}=\rho dV\]

 

\[dV=(dl_{n})(dA)\]

\[dl_{n}=(dl)(-cos\theta)\]

\[dl=(\widetilde{V})(\Delta t)\]

\[dV=(\widetilde{V})(\Delta t)(-cos\theta)(dA)\]

 

\[dB_{in}=bdm_{in}\]

\[dB_{in}=b\rho(\widetilde{V})(\Delta t)(-cos\theta)(dA)\]

 

\[\int_{CS_{in}}^{}dB_{in}=B_{I}(t+\Delta t)\]

\[\int_{CS_{in}}^{}dB_{in}=\int_{CS_{in}}^{}b\rho(\widetilde{V})(\Delta t)(-cos\theta)(dA)\]

 

\[\lim_{\Delta t\rightarrow 0} \frac{B_{I}(t+\Delta t) }{\Delta t}\]

\[\equiv \dot{B}_{in}\]

 

\[\lim_{\Delta t\rightarrow 0} \frac{B_{I}(t+\Delta t) }{\Delta t}\]

\[=\lim_{\Delta t\rightarrow 0} \frac{1}{\Delta t}\left (\int_{CS_{in}}^{}b\rho(\widetilde{V})(\Delta t)(-cos\theta)(dA) \right )\]

\[=\int_{CS_{in}}^{}b\rho(\widetilde{V})(-cos\theta)(dA) \]

 

\[\widetilde{V}(cos \theta)=\vec{\widetilde{V}}\cdot \vec{n}\]

 

\[\dot{B}_{in}=\lim_{\Delta t\rightarrow 0} \frac{B_{I}(t+\Delta t) }{\Delta t}\]

\[=-\int_{CS_{in}}^{}b\rho\vec{\widetilde{V}}\cdot \vec{n}(dA) \]

 

 

\[\frac{dB_{sys}}{dt}=\frac{dB_{CV}}{dt}+\dot{B}_{out}-\dot{B}_{in}\]

\[\frac{dB_{sys}}{dt}=\frac{d}{dt} \int_{CV}^{} \rho b dV+\int_{CS_{out}}^{}b\rho\vec{\widetilde{V}}\cdot \vec{n}(dA)-\left (-\int_{CS_{in}}^{}b\rho\vec{\widetilde{V}}\cdot \vec{n}(dA)\right )\]

\[\frac{dB_{sys}}{dt}=\frac{d}{dt} \int_{CV}^{} \rho b dV+\int_{CS}^{}b\rho\vec{\widetilde{V}}\cdot \vec{n}(dA)\]

 

B=m 인 경우

질량보존법칙은 아래와 같다.

\[0=\frac{dm_{sys}}{dt}=\frac{dm_{CV}}{dt}+\dot{m}_{out}-\dot{m}_{in}\]

\[\frac{dm_{CV}}{dt}=\dot{m}_{in}-\dot{m}_{out}\]

 

B=E 인 경우

에너지보존법칙(열역학 제1법칙)은 아래와 같다.

\[0=\frac{dE_{sys}}{dt}=\frac{dE_{CV}}{dt}+\dot{E}_{out}-\dot{E}_{in}\]

\[\frac{dE_{CV}}{dt}=\dot{E}_{in}-\dot{E}_{out}\]

 

 

 

 

 

 

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[1] Cengel, Yuns A.; Cimbala, John M. (2018). Flouid Dynamics: Fundamentals and Applications (4th ed.). McGraw-Hill. pp. 164-171.

[2] Gerhart, Philip M.; Gerhart, Andrew L.; Hochstein John I. (2018). Munson, Young, and Okiishi's Fundamentals of Fluid Mechanics (8th ed.). Wiley. pp. 175-187.