Essential Formulae

J. Gordon Leishman

Essential Formulae

Note: All symbols have their usual meanings as used in the text. These essential formulas may be made available for use on quizzes and exams with the permission of the instructor.

Structural stress relationships:

$\sigma = \frac{T}{A}$

$\epsilon = E \sigma$

Equation of state:

$p = \varrho R T$

Hydrostatic pressure in a stagnant fluid:

$\nabla p = \varrho \vec{f}_{b} -\varrho g\, \vec{k}$

Hydrostatic pressure in scalar form:

$\begin{eqnarray*} \frac{\partial p}{\partial x} & = & \varrho f_x \nonumber \\[18pt] \frac{\partial p}{\partial y} & = & \varrho f_y \nonumber \\[18pt] \frac{\partial p}{\partial z} & = & \varrho f_z -\varrho g \nonumber \end{eqnarray*}$

Hydrostatic equation:

$\frac{dp}{dz} = -\varrho g$

Unique solutions to the hydrostatic equation:
- For a constant-density fluid:

$p + \varrho g z = \mbox{constant}$

- Pressure change with a linear thermal gradient with temperature lapse $B$ between two heights $z_1$ and $z_2$ :

$\frac{p_2}{p_1} = \left|\frac{T_0 - \alpha z_2}{T_0 - \alpha z_1}\right|^{g /R \alpha}$

- Pressure change with an isothermal temperature $T_0$ between two heights $z_1$ and $z_2$ :

$\frac{p_2}{p_1} = \exp\bigg( \frac{ -g (z_2 - z_1)}{R T_0}\bigg)$

Specific gravity SG of a liquid = $\varrho/\varrho_{\rm H_2 0}$

$\[\]$

Specific weight $\gamma$ of a liquid = $\varrho g$

$\[\]$

Temperature conversions:
- Convert temperature $T$ to Kelvin K from Centigrade or Celsius $^{\circ}\mbox{C}$ :

$\mbox{K} = \, ^{\circ}\mbox{C} + 273.15$

- Convert temperature $T$ to Rankine R from Fahrenheit $^{\circ}\mbox{F}$ :

$\mbox{R} = \, ^{\circ}\mbox{F} + 459.67$

Speed of sound:

$a = \sqrt{ \gamma R T}$

Mach number:

$M = \frac{V}{a}$

Reynolds number based on length $l$ :

$Re = \frac{\varrho V l}{\mu}$

Kinematic viscosity = $\mu/\varrho$

Shear stress in a Newtonian fluid:

$\tau=\mu \left( \frac{du}{dy} \right)$

Equations of a streamline:

$\begin{gather*} d\vec{l} \times \vec{V} = \vec{0} \\[12pt] \frac{dy}{dz}=\frac{v}{w},~~\frac{dz}{dx}=\frac{w}{u},~~\frac{dy}{dx}=\frac{v}{u} \end{gather*}$

Continuity equation for a fluid:

$\frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, d {\cal{V}} + \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} = 0$

Momentum equation for a fluid:

$\vec{F} = \oiiint_{\cal{V}} \varrho \vec{f} _b d{\cal{V}} -\oiint_S p dS + \vec{F}_{\mu} = \frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, \vec{V} d {\cal{V}} + \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V}$

The energy equation for a fluid:

$\begin{eqnarray*} \oiiint_{\cal{V}} \overbigdot{q} \varrho \, d {\cal{V}} + Q_{\mu} -\oiint_{S} (p d\vec{S}) \bigcdot \vec{V} + \oiiint_{\cal{V}} ( \varrho \vec{f}_b d{\cal{V}}) \bigcdot \vec{V} + \overbigdot{W}_{\mu} + \overbigdot{W}_{\rm mech} = && \nonumber \\ \frac{\partial}{\partial t} \oiiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}} + \oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z\right) & \nonumber \end{eqnarray*}$

The energy equation for the single-stream system:

$w_{\rm mech} = \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1 ) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right)$

Bernoulli equation:

$p + \frac{1}{2} \varrho V^2 + \varrho g z = \mbox{constant}$

The hydraulic diameter of a pipe:

$\begin{equation*} D_h = \frac{4 \times \mbox{Area~of~cross-section}}{\mbox{Perimeter}} = \frac{4 A_c}{p} \end{equation*}$

Pressure drop $\Delta p_L$ for a pipe flow along a uniform diameter pipe of length $L$ :

$\begin{equation*} \Delta p_L = \frac{1}{2} \varrho V_{\rm av}^2 \, f \left( \frac{L}{D} \right) \end{equation*}$

Head loss for a pipe flow:

$h_L = \frac{\Delta p}{\varrho \, g}$

Pumping power for a pipe flow:

$P = Q \, \Delta p$

Area of a wing:

$\begin{equation*} S = 2 \int_{0}^{s} c \, dy = 2 \int_{0}^{b/2} c \, dy \end{equation*}$

The aspect ratio of a wing:

$\begin{equation*} A\!R = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{2 \displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}$

Mean aerodynamic chord (MAC) of a wing:

$\begin{equation*} {\rm MAC} = \overline{\overline{c}} = \frac{2 \displaystyle{\int_{0}^{s} c^2 dy}}{S} \end{equation*}$

Free stream dynamic pressure:

$q_{\infty} = \frac{1}{2} \varrho_{\infty} \, V_{\infty} ^2$

Aerodynamic coefficients for an airfoil section:
- Lift coefficient, $C_{l} = \displaystyle{\frac{L/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{L'}{q_{\infty} c}}$

- Drag coefficient, $C_{d} = \displaystyle{\frac{D/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{D'}{q_{\infty} c}}$

- Moment coefficient at some point $a$ , $C_{m_{a}} = \displaystyle{\frac{M_a/ \mbox{unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_a}{q_{\infty} c^2}}$
Aerodynamic coefficients for a finite wing:
- Lift coefficient, $C_{L} = \displaystyle{\frac{L}{q_{\infty} S}}$

- Drag coefficient, $C_{D} = \displaystyle{\frac{D}{q_{\infty} S}}$

- Moment coefficient about some point $a$ , $C_{M_{a}} = \displaystyle{\frac{M_a}{q_{\infty} S \overline{\overline{c}}}}$

Induced drag coefficient of a finite wing:

$\begin{equation*} C_{D_{i}} = \frac{(1 + \delta) {C_L}^2}{\pi \, A\!R} \end{equation*}$

The drag coefficient for an airplane:

$\begin{equation*} C_D = C_{D_{0}} + \frac{ {C_L}^2}{\pi \, AR \, e} = C_{D_{0}} + k {C_L}^2 \end{equation*}$

General equations of motion of an airplane:

$\begin{eqnarray*} \mbox{$\parallel$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{d V_{\infty}}{dt} = T \cos \epsilon - D - W \sin \theta \\[18pt] \mbox{$\perp$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{V_{\infty}^2}{r_1} = L \cos \phi + T \sin \epsilon \cos \phi - W \cos \theta \\[18pt] \mbox{Horizontal plane:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{(V_{\infty} \cos \theta)^2}{r_2} = L \sin \phi + T \sin \epsilon \sin \phi \end{eqnarray*}$

Breguet endurance and range equations for a propeller aircraft:

$E = \frac{ \eta_p}{{\rm BSFC}} \left(\frac{C_L^{\, 3/2}}{C_D}\right) \sqrt{2 \varrho S} \left( \frac{1}{\sqrt{W_0 - W_f}} - \frac{1}{\sqrt{W_0} } \right)$

$R = \frac{\eta_p}{{\rm BSFC}} \left( \frac{C_L}{C_D}\right) \ln \left( \frac{W_0}{W_0-W_f}\right)$

Breguet endurance and range equations for a jet aircraft:

$E = \frac{1}{{\rm TSFC}} \left( \frac{C_L}{C_D} \right) \ln \left(\frac{W_0}{W_0 - W_f} \right)$

$R = \frac{2}{{\rm TSFC}} \sqrt{ \frac{2}{\varrho S}} \left(\frac{C_L^{1/2}}{C_D} \right) \left( W_0^{1/2} - W_1^{1/2} \right)$

The equivalent exit velocity of a rocket in terms of specific impulse:

$V_{\rm eq} = I_{\rm sp} \, g_0$

Thrust equation for rocket:

$T = \overbigdot{m} \, V_{\rm eq}$

Rocket equation (with gravity loss term):

$\Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - g_0 t_b$

Mean equivalent exhaust velocity (for multiple boosters):

$\overline{V}_{\rm eq} = \frac{ N_b \, \overbigdot{m}_b \, V_{{\rm eq}_{b}} + \overbigdot{m}_1 \, V_{{{\rm eq}_{1}}} }{ N_b \overbigdot{m}_b + \overbigdot{m}_1 }$

Conversion factors:
- 1 knot = 1.68781 ft/s
- 1 knot = 0.51444 m/s
- 1 mph = 0.44704 m/s
- 1 mph = 1.46667 ft/s
- 1 ft = 0.3048 m
- 1 statute mile = 5,280 ft
- 1 nautical mile = 6,076 ft
- 1 hp = 550 ft-lb s $^{-1}$