## Network Modeling | | Mechanical | Electrical | Fluidic | Thermal | | ----------- | :----------------: | :-----------------------: | :-------------------------------: | :-------------------------------: | | Unit | $x$, [m] | $Q$, [C] | $m$, [kg] | $Q$, [J] | | Potential | $F$, [N] | $V$, [V] | $\Delta P$ [Pa] | $\Delta T$ [K] | | Flux | $v$, [m/s] | $I$, [A] | ${} q$, [$\text{m}^{3}/\text{s}$] | ${} \dot{Q} {}$, [W] | | Inertia | ${} F=m\dot{v} {}$ | ${} V=L \frac{di}{dt} {}$ | ${} \Delta P=L \frac{dq}{dt} {}$ | N/A | | Resistance | ${} F=bv {}$ | ${} V=iR {}$ | ${} \Delta P=qR {}$ | ${} \Delta T = R\cdot \dot{Q} {}$ | | Capacitance | $F=kx$ | ${} i=C \frac{dV}{dt} {}$ | ${} m = C\Delta P {}$ | $Q=mc\Delta T$ | | Power | $P = F\cdot v$ | ${} P = V\cdot i {}$ | ${} P = \Delta P \cdot q {}$ | $P = \dot{Q}$ | ## Conservation Laws ### Energy $ \frac{dE_{\text{cv}}}{dt}=\dot{Q}-\dot{W}+\sum_{\text{in}}\dot{m}_{\text{in}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{in}} -\sum_{\text{out}}\dot{m}_{\text{out}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{out}} $ $h = u + \frac{P}{\rho}$ ### Linear Momentum $ \sum \vec{F} = m\vec{a} $ For an inertial control volume: $ \sum \vec{F} = \frac{d}{dt} \int_{CV}\vec{v} \rho dV + \sum_{out}\dot{m}_{i}\vec{v}_{i} - \sum_{in} \dot{m}_{i} \vec{v}_{i} $ **Navier Stokes** $\rho \frac{D\vec{v}}{Dt} = -\vec{\nabla}P + \rho \vec{g} + \mu \nabla^{2} \vec{v}$ **Material Derivative** $\frac{D}{Dt} = \frac{\delta}{\delta t} + \vec{\nabla} \cdot \vec{v}$ ### Angular Momentum $\sum \vec{\tau}=\frac{d}{dt}(r\times mv)=I\dot{\omega}$ ### Entropy $ \frac{dS_{\text{cv}}}{dt}=\frac{\dot{Q}}{T_{\text{s}}}+\dot{S}_{\text{gen}}+\sum_{\text{in}} \dot{m}_{\text{in}}S_{\text{in}} - \sum_{\text{out}} \dot{m}_{\text{out}} S_{\text{out}} $