Method to switch from ***system*** analysis to ***control volume*** analysis - System: collection of matter of fixed identity - same set of particles - specific quantity of matter - Control volume: a fixed volume in space that particles pass through - independent of mass General form: $ \frac{D\phi_{sys}}{Dt}=\underbrace{\frac{\partial}{\partial t}\int_{CV}\phi\rho\, dV}_{\substack{\text{rate of change of }\phi \\ \text{ in CV}}}+\underbrace{\int_{CS}\phi\rho (\vec{u}\cdot\vec{n})\,dA}_{\substack{\text{net flux of }\phi\\\text{ across CS}}}=0 $ Conservation of mass: $ \frac{Dm_{sys}}{Dt}=\frac{\partial}{\partial t}\int_{CV}\rho dV+\int_{CS}\rho (\vec{u}\cdot\vec{n})dA=0 $ Conservation of momentum: - inertial reference frame $ \Sigma \vec{F}=\frac{\partial}{\partial t}\int_{CV}\vec{u}\rho dV+\int_{CS}\vec{u}\rho(\vec{u}\cdot\vec{n})dA $ - non-inertial reference frame $xyz$ $ \Sigma \vec{F}-\int_{CV}\vec{a}_{rf}\rho dV=\frac{\partial}{\partial t}\int_{CV}\vec{U}_{xyz}\rho dV+\int_{CS}\vec{U}_{xyz}\rho(\vec{U}_{xyz}\cdot\vec{n})dA $ Conservation of energy: $ \dot{Q}-\dot{W}_{sh}-\dot{W}_{vs}=\frac{\partial}{\partial t}\int_{CV}(\tilde{u}+u^2/2+gz)\rho dV+\int_{CS}(h+u^2/2+gz)\rho(\vec{u}\cdot\vec{n})dA $