## 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 | $F=kx$ | ${} i=C \frac{dV}{dt} {}$ | ${} Q = C \cdot\dot{P} {}$ | ${} \dot{Q}=mc\Delta T {}$ | | 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 ## Math ### Taylor Series $ f(x + h) = f(x) + f'(x)h + f''(x) \frac{h^{2}}{2!} + \dots $ ### Material Derivative $ \frac{D}{Dt} = \left( \frac{\delta}{\delta t} + v_{x} \frac{\delta}{\delta x} + v_{y} \frac{\delta}{\delta y} + v_{z} \frac{\delta}{\delta z} \right) = \frac{\delta}{\delta t} + (\vec{v} \cdot \vec{\nabla}) $ ## Material Properties ### Solids | | **$E$** | **$\nu$** | **$\rho$** | **$F_{ty}$** | **$F_{tu}$** | **$\alpha$** | **$c$** | **$k$** | | ------------- | ---------------: | --------: | -----------------------: | ---------------: | ---------------: | -----------: | -----------------------: | ----------------------: | | | *$\mathrm{GPa}$* | | *$\mathrm{ kg / m^{3}}$* | *$\mathrm{MPa}$* | *$\mathrm{MPa}$* | | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{W / (m ~K)}$* | | Al 6061 | 68 | 0.33 | 2700 | 240 | 290 | 23.6 | 896 | 167 | | Al 7075 | 71 | 0.33 | 2800 | 345 | 415 | 23.5 | 960 | 130 | | Steel 4130 | 200 | 0.32 | 7850 | 517 | 655 | 12.2 | 477 | 42.7 | | Stainless 304 | 193 | 0.30 | 8000 | 215 | 505 | 17.2 | 500 | 16 | | Ti‑6Al‑4V | 114 | 0.34 | 4430 | 880 | 950 | 8.6 | 526 | 6.7 | | ABS | 2.1 | 0.35 | 1040 | 46 | 65 | 80 | 1350 | 0.17 | | Nylon 6 | 2.8 | 0.39 | 1140 | 70 | 95 | 80 | 1700 | 0.25 | | Polycarbonate | 2.3 | 0.37 | 1200 | 65 | 70 | 65 | 1200 | 0.20 | ### Fluids | | **$\rho$** | **$c_{p}$** | **$c_{v}$** | **$\mu$** | **$k$** | **$\mathrm{Pr}$** | | ------------ | -----------------------: | -----------------------: | -----------------------: | ----------------------: | ----------------------------: | ----------------: | | | *$\mathrm{ kg / m^{3}}$* | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{Pa \cdot s}$* | *${} \mathrm{W / (m ~K)} {}$* | | | Air (20°C) | 1.2 | 1005 | 718 | 0.0000181 | 0.026 | 0.71 | | Water (20°C) | 1000 | 4182 | 4175 | 0.001 | 0.60 | 7.0 | | Oil | 900 | 2000 | 1800 | 0.1 | 0.15 | 100 |