User:Cirrus246/sandbox

--- This page is only used to create equations for other purposes off wikipedia. ---


 * $$ C_{L_\alpha} = 2 \pi { AR \over AR + 2 }$$

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 * $$ Cp = 4a (1 - a)^2 $$


 * $$ Cp_{max} = 16/27 (=0.59259) \; \hbox{when a = 1/3} $$

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 * $$ \dot\theta$$


 * $$ \ddot\theta$$


 * $$ k = {f c \over V} $$


 * $$f = \dot \delta_{\;\hbox{flap}} = 10 \; \hbox{deg/sec} = 0.175 \; \hbox{rad/sec}$$


 * $$ c_{\;\hbox{flap chord at 0.75R}} = c \, (c_f / c) = (4 \; \hbox{m}) \times 0.2 = 0.8\; \hbox{m} $$


 * $$ V = V_{\;\hbox{avg}} \; TSR = (9.08 \; \hbox{m/s} ) \times 9.25 = 83.99 \; \hbox{m/s}$$


 * $$ k = 0.175 \times 0.8 / 83.99 = 0.0017 \;\;\hbox{based on flap rate}$$


 * $$ R_2 = \sqrt{ {P_2 \over P_1} R_1^2}$$


 * $$ L = q S C_L$$


 * $$ Re = { V_{\infty} c \over \nu }$$


 * $$ V_{a} = a V_{\infty}$$


 * $$ V_{tip,cond} = V_{wind,cond} TSR $$


 * $$ TSR = V_{tip,cond} / V_{wind,cond} $$


 * $$ TSR = \Omega R / V_{wind,cond} $$


 * $$ \overline{V}_{H} = { l_{H} S_{H} \over c_W S_W  }$$


 * $$ l_{H} S_{H} = \overline{V}_{H} c_W S_W$$


 * $$ \overline{V}_{V} = { l_{V} S_{V} \over b_W S_W  }$$


 * $$ l_{V} S_{V} = \overline{V}_{V} b_W S_W$$


 * $$ S = bc \;\, \& \;\, AR = b/c $$


 * $$ b = \sqrt{S \;AR} \;\,\, \& \;\, c = b/AR $$


 * $$ \lambda = c_{tip} / c_{root} \;\,\, \& \;\, c = (c_{tip} + c_{root})/2$$


 * $$ c_{root} = 2 c / (1 + \lambda) \;\, \& \;\, c_{tip} = \lambda\, c_{root} $$


 * $$ c_{avg, H} = {b_H \over AR_H} $$


 * $$ c_{avg, V} = {b_V \over AR_V} $$

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 * $$ P \propto W ^{3/2}  $$


 * $$ P = { W \sqrt{ 2 (W/S) \over \rho}   \over  C_L^{3/2} / C_D      } $$


 * $$ C_D = C_{d_{airfoil}} + C_{D_{0,fuse}} + C_{D_{0,tail}} + C_{D,i}$$


 * $$ C_{D,i} = {C_L^{\,2}  \over \pi e AR}$$

Turbulent flow

 * $$C_{f} = \frac{0.455}{[\log_{10}(\mathrm{Re})]^{2.58}} \ $$              Also known as the Schlichting empirical formula


 * $$ D_{sf} = q S_w C_{f} $$


 * $$ R = \sqrt{2 (W/S) \over \rho} {c \over \nu}$$


 * $$ V = \sqrt{2 (W/S) \over \rho\, C_L}$$


 * $$ Re = {V c \over \nu}$$


 * $$ R = Re \sqrt{ C_L }$$


 * $$ Re = {R \over \sqrt{ C_L }}$$

Take the axial induction factor to be 1/3 for best $$C_P$$. Assume that $$C_P$$ is close to ideal (59%), say, $$C_P = 0.50$$. The power coefficient is given by:


 * $$C_P = {P \over \rho V^3 (\pi R^2) / 2} $$

Rearrange to show that:


 * $$R = \sqrt{P \over \pi \rho V^3 C_P / 2}$$

Consider the rated power condition. For a fixed rated power and rated wind speed, the radius is the same. The radius is then independent of the number of blades having assumed $$C_P$$ to be constant.

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 * $$w_{axial}^\prime(r) = \sqrt{2}\, w_{axial}(r) = 1.41\,w_{axial}(r) $$


 * $$ w_{avg} A = \sum 2 \pi r w dr$$


 * $$ w_{avg} = {\sum 2 \pi r w dr \over A } = {\sum 2 \pi r w dr \over \pi R^2 } $$


 * $$ w_{avg} = {\sum 2 \pi r w dr \over \pi } $$


 * $$ R/C = {P_x \over W}  \sin{\gamma} = {T_x \over W} $$


 * $$ V_v = V sin \overline{\gamma} = V { D \over W} = {P_R \over W} $$


 * $$ V_{v, min} = {P_{R, min} \over W} $$

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 * $$ b\, (\rho V^2 / 2) c\, C_l \sim L'$$


 * $$ C_L = L / q S = { L \over \frac{1}{2} \rho V^2 S}$$


 * $$ V \sim \Omega r_{82.5\%}/R$$


 * $$ b\, \Omega^2 c\, C_l = K_2 $$


 * $$ b\, \Omega^2 c\, C_l = 1,379.771 $$


 * $$ [\Omega] = \mbox{rad/sec, } [c] = \mbox{in}$$


 * $$ Re \sim V c$$


 * $$ V \sim \Omega$$


 * $$ \Omega c \sim {1 \over b\, C_l\, \Omega}$$


 * $$ Re \sim {K_1 \over b\, C_l\, \Omega}$$


 * $$ Re = {7,199,059 \over b\, C_l\, \Omega}$$


 * $$ [\Omega] = \mbox{rad/sec, } [b] = \mbox{number of blades}$$

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https://en.wikipedia.org/wiki/Elo_rating_system


 * $$ \mbox{Guess initial propeller speed } \hat{n}$$


 * $$ \mbox{Step 1: } D, V, \hat{n} \rightarrow \hat{J}$$


 * $$ \mbox{Step 2: With } \hat{J}, \mbox{interpolate to get the estimate } \hat{C}_T$$


 * $$ \mbox{Step 3: With } \hat{C}_T, \hat{n}, D, \rho \rightarrow \hat{T}$$


 * $$ \mbox{Step 4: } \hat{T} = T_{spec} ? \mbox{ If `yes', finished. Otherwise, continue. } $$


 * $$ \mbox{Step 5: Estimate new guess for } \hat{n} \mbox{ using }\hat{n}_{new} = \hat{n}_{old} + \omega \Delta n \mbox{ where } \Delta n = {\Delta T \over 2 \rho n D^4 C_T}$$


 * $$ \mbox{where } \Delta T = T_{spec} - \hat{T}$$


 * $$ \mbox{and } \Delta n \mbox{ derives from }$$


 * $$T = \rho n^2 D^4 C_T$$


 * $${dT \over dn} = 2 \rho n D^4 C_T$$


 * $${\Delta T \over \Delta n} = 2 \rho n D^4 C_T$$


 * $$\Delta n = {\Delta T \over 2 \rho n D^4 C_T}$$


 * $$ \mbox{and } \omega \mbox{ is a relaxation factor (= 0.5). }$$


 * $$ \mbox{Step 6: Repeat by going to Step 1.}$$

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 * $$ \mbox{This is regular text } x = y $$
 * $$J = {V \over nD }$$


 * $$C_T = {T \over \rho n^2 D^4 }$$


 * $$C_P = {P \over \rho n^3 D^5 }$$


 * $$C_Q = {Q \over \rho n^2 D^5 }$$


 * $$\eta = {T V \over P}$$


 * $$\eta = {C_T J \over C_P}$$


 * $$\eta = {C_T J \over 2 \pi C_Q}$$


 * $$C_P = 2 \pi C_Q$$


 * $$C_Q = {C_P \over 2 \pi}$$


 * $$C_Q = {C_T J \over 2 \pi \eta}$$


 * $$n = \mbox{revolutions per sec}$$

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 * $$J = {V \over nD }$$


 * $$C_T = {T \over \rho n^2 D^4 }$$


 * $$C_P = {P \over \rho n^3 D^5 }$$


 * $$C_Q = {Q \over \rho n^2 D^5 }$$


 * $$w_0 = \sqrt{T \over 2 \rho A }$$


 * $$A = \pi R^2$$


 * $$CFM = A w_0$$

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 * $$CFM = A w_0 = \sqrt{A T \over 2 \rho }$$


 * $$CFM^\prime = \sqrt{A T \over \rho } = \sqrt{2}\, CFM = 1.41\, CFM $$