Essential Formulae for AE201 Students

J. Gordon Leishman

Essential Formulae for AE201 Students

Note: All symbols have their usual meanings as used in the eBook. These essential formulas may be used on AE201 quizzes and exams with the instructor’s permission; in that case, an official formula sheet will be provided.

Structural stress/strain relationships:

$\sigma = \frac{T}{A} \quad \quad \epsilon = \frac{\sigma}{E}$

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 temperature lapse rate $\alpha$ between two heights ${z_{1}}$ and ${z_2}$ :

$\frac{p_2}{p_1} = \left(\frac{T_0 - \alpha z_2}{T_0 - \alpha z_1}\right)^{\tfrac{g}{R\alpha}}$

- 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 = $\varrho/\varrho_{\rm H_2O}$

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:

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

Mass flow rate for one-dimensional isentropic compressible flow:

$\overbigdot{m} = A\, \frac{p_0}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}}\, M \left( 1+\frac{\gamma-1}{2}M^2 \right)^{-\tfrac{\gamma+1}{2(\gamma-1)}}$

Choked mass flow rate:

$\overbigdot{m}_{\max} = A^*\, \frac{p_0}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} \left( \frac{2}{\gamma+1} \right)^{\tfrac{\gamma+1}{2(\gamma-1)}}$

Critical pressure ratio at the sonic condition:

$\frac{p^*}{p_0} = \left( \frac{2}{\gamma+1} \right)^{\tfrac{\gamma}{\gamma-1}}$

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:

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

Continuity equation in 1-D form:

$\sum_{S}\varrho \, A \, V = \overbigdot{m} = \text{constant}$

Momentum equation for a fluid:

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

Momentum equation in 1-D form:

$F_{x} = - \left( p_{2} A_{2} - p_{1} A_{1} \right) + \overbigdot{m} \left( u_2 - u_1 \right)$

The energy equation for a fluid:

$\begin{aligned} &\oiiint_{\cal{V}} \overbigdot{q}\,\varrho\,d{\cal{V}} + Q_{\mu} - \oiint_S \left(p\,d\vec{S}\right)\bigcdot\vec{V} + \oiiint_{\cal{V}} \varrho\vec{f}_b\bigcdot\vec{V}\,d{\cal{V}} + \overbigdot{W}_{\mu} + \overbigdot{W}_{\rm mech} \\[6pt] &\qquad = \frac{\partial}{\partial t} \oiiint_{\cal{V}} \varrho \left( e+\frac{V^2}{2}+gz \right)d{\cal{V}} + \oiint_S \left( \varrho\vec{V}\bigcdot d\vec{S} \right) \left( e+\frac{V^2}{2}+gz \right) \end{aligned}$

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:

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

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:

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

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:

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

Lift and drag:

$L = q_{\infty} S C_L \qquad D = q_{\infty} S C_D$

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:

$C_{D_{i}} = \frac{(1 + \delta) {C_L}^2}{\pi \, AR}$

The drag coefficient for an airplane:

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

Stall speed:

$V_s = \sqrt{\frac{2W}{\varrho S C_{L_{\max}}}}$

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*}$

Load factor:

$n = \frac{L}{W}$

Coordinated level turn:

$n = \frac{1}{\cos\phi} \qquad r = \frac{V^2}{g\tan\phi}$

Thrust and power required:

$T_R = D \qquad P_R = D V$

Rate of climb:

$V_c = \frac{P_A-P_R}{W}$

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:

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

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

Thrust produced by a turbojet:

$T = \left(\overbigdot{m}_{\rm air} + \overbigdot{m}_{\rm f} \right) V_{j} - \overbigdot{m}_{\rm air} V_{\infty}$

Thrust specific fuel consumption (TSFC):

${\rm TSFC} = c_t = \frac{\overbigdot{W}_{\rm f}}{T}$

Brake power specific fuel consumption (BSFC):

${\rm BSFC} = c_b = \frac{\overbigdot{W}_{\rm f}}{P}$

Propeller efficiency:

$\eta_p = \frac{T V}{P_{\rm shaft}}$

Propulsive efficiency for a jet-producing device:

$\eta_{P} = \frac{2}{1+\left(\dfrac{V_{j}}{V_{\infty}}\right)}$

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 $^{-1}$