Fluid Mechanics
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Darcy-Weisbach Equation (Pipe Head Loss)
$$\Delta h_f = f \cdot \frac{L}{D} \cdot \frac{u^2}{2g} \quad \Longleftrightarrow \quad \Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho u^2}{2}$$
Variables: $f$ = Darcy friction factor, $L$ = pipe length ($\text{m}$), $D$ = internal diameter ($\text{m}$), $u$ = mean flow velocity ($\text{m/s}$), $\rho$ = fluid density ($\text{kg/m}^3$), $g = 9.80665\text{ m/s}^2$.
Valid for all steady incompressible pipe flows (laminar and turbulent).
Fluid Mechanics
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Colebrook-White Implicit Equation (Friction Factor)
$$\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\epsilon / D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right)$$
Variables: $f$ = Darcy friction factor, $\epsilon$ = equivalent wall surface roughness ($\text{m}$), $D$ = pipe diameter ($\text{m}$), $Re = \rho u D / \mu$ = Reynolds number ($Re > 4000$).
Solved iteratively via Newton-Raphson or pure Fortran solver.
Heat Conduction
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Fourier's Law of Heat Conduction (1D Steady)
$$q'' = -k \frac{dT}{dx} \quad \Longrightarrow \quad \dot{Q} = \frac{T_1 - T_2}{R_{th}} = \frac{k A}{L} (T_1 - T_2)$$
Variables: $\dot{Q}$ = heat transfer rate ($\text{W}$), $q''$ = heat flux ($\text{W/m}^2$), $k$ = thermal conductivity ($\text{W/(m}\cdot\text{K)}$), $A$ = surface area ($\text{m}^2$), $R_{th} = L / (kA)$ = thermal resistance ($\text{K/W}$).
Fundamental equation governing conductive heat diffusion in solids.
Heat Exchangers
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Log Mean Temperature Difference (LMTD)
$$\dot{Q} = U A F \cdot \Delta T_{lm} \quad \text{where} \quad \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$$
Variables: $U$ = overall heat transfer coefficient ($\text{W/(m}^2\cdot\text{K)}$), $A$ = heat exchange area ($\text{m}^2$), $F$ = geometry correction factor ($F \le 1.0$), $\Delta T_1, \Delta T_2$ = temperature differences at exchanger inlet and outlet.
Counter-flow exchangers always yield higher $\Delta T_{lm}$ than parallel flow.
Heat Exchangers
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Effectiveness-NTU ($\epsilon$-NTU) Relationship
$$\epsilon = \frac{\dot{Q}}{\dot{Q}_{max}} = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r \exp[-NTU(1 - C_r)]} \quad \left(NTU = \frac{U A}{C_{min}}, \; C_r = \frac{C_{min}}{C_{max}}\right)$$
Variables: $\epsilon$ = thermal effectiveness ($0 \le \epsilon \le 1$), $C_{min}, C_{max}$ = minimum and maximum heat capacity rates ($\dot{m} c_p$), $NTU$ = Number of Transfer Units.
Direct method for evaluating exchanger performance without knowing outlet temperatures.
Thermal Radiation
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Stefan-Boltzmann Thermal Radiation Law
$$E_b = \sigma T^4 \quad \Longrightarrow \quad \dot{Q}_{12} = \epsilon \sigma A (T_1^4 - T_2^4)$$
Variables: $\sigma = 5.670374 \times 10^{-8}\text{ W/(m}^2\cdot\text{K}^4)$ (Stefan-Boltzmann constant), $\epsilon$ = surface emissivity ($0 < \epsilon \le 1$), $T$ = absolute temperature in Kelvin ($\text{K}$).
Non-linear $T^4$ dependence dominates at elevated temperatures.
Compressible Aerodynamics
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Rankine-Hugoniot Normal Shock Relations
$$M_2^2 = \frac{2 + (\gamma - 1)M_1^2}{2\gamma M_1^2 - (\gamma - 1)}, \quad \frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2 - 1)$$
Variables: $M_1$ = upstream Mach number ($M_1 > 1$), $M_2$ = downstream Mach number ($M_2 < 1$), $\gamma = c_p/c_v$ = specific heat ratio (1.4 for air), $P_2/P_1$ = static pressure jump across shock.
Exact analytical jump conditions across a 1D normal shock wave.
Compressible Aerodynamics
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Oblique Shock $\theta$-$\beta$-$M$ Relation
$$\tan\theta = 2 \cot\beta \left[ \frac{M_1^2 \sin^2\beta - 1}{M_1^2 (\gamma + \cos 2\beta) + 2} \right]$$
Variables: $\theta$ = flow deflection / wedge half-angle, $\beta$ = shock wave angle, $M_1$ = upstream Mach number, $\gamma$ = specific heat ratio.
Yields weak shock (supersonic downstream) and strong shock (subsonic) roots.
Compressible Aerodynamics
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Prandtl-Meyer Expansion Function $\nu(M)$
$$\nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \arctan \sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2 - 1)} - \arctan \sqrt{M^2 - 1}$$
Variables: $\nu(M)$ = Prandtl-Meyer angle ($\text{rad}$), $\theta = \nu(M_2) - \nu(M_1)$ = convex corner turning angle.
Isentropic supersonic expansion around convex corners and airfoils.
Mass Transfer
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Fick's First Law of Molecular Diffusion
$$J_A^* = -D_{AB} \frac{dC_A}{dz} \quad \Longleftrightarrow \quad j_A = -\rho D_{AB} \frac{dw_A}{dz}$$
Variables: $J_A^*$ = molar diffusion flux ($\text{mol/(m}^2\cdot\text{s)}$), $D_{AB}$ = binary diffusion coefficient ($\text{m}^2\text{/s}$), $C_A$ = molar concentration ($\text{mol/m}^3$).
Analogous to Fourier's law of conduction ($k \leftrightarrow D_{AB}, T \leftrightarrow C_A$).