## Network Modeling | | Mechanical | Electrical | Fluidic | Thermal | | ----------- | :----------------: | :-----------------------: | :----------------------------: | :-------------------------------: | | Unit | $x$, [m] | $q$, [C] | $V$, [$\text{m}^3$] | $Q$, [J] | | Potential | $F$, [N] | $V$, [V] | $P$ [Pa] | $\Delta T$ [K] | | Flux | $v$, [m/s] | $I$, [A] | $Q$, [$\text{m}^{3}/\text{s}$] | $\dot{Q}$, [J/s] | | Inductance | ${} F=m\dot{v} {}$ | ${} V=L \frac{di}{dt} {}$ | $P=L \frac{dq}{dt}$ | N/A | | Resistance | ${} F=bv {}$ | ${} V=iR {}$ | $P=R\cdot Q$ | ${} \Delta T = R\cdot \dot{Q} {}$ | | Capacitance | $F=kx$ | ${} i=C \frac{dV}{dt} {}$ | ${} Q = C \cdot\dot{P} {}$ | $Q=mc\Delta T$ | | Power | $P = F\cdot v$ | ${} P = V\cdot i {}$ | ${} \dot{W} = P \cdot Q {}$ | $P = \dot{Q}$ | ## 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 + \left( \sum\dot{m}\vec{v} \right)_\text{out} - \left( \sum \dot{m} \vec{v} \right)_{\text{in}} $ #### 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}}+\left( \sum \dot{m}S \right)_{\text{in}} - \left( \sum \dot{m} S \right)_\text{out} $ $\dot{S}_{\text{gen}}>0$ always