Let's break down and solve the problem step by step: --- ## **Problem Restatement** We have: - **Parameters:** air density (\(\rho\)), wind speed (\(V\)), rotor radius (\(R\)), rotor angular speed (\(\omega\)). - **Prototype (Full Scale):** \(R_p = 50~\text{m}\), \(V_p = 5~\text{m/s}\), \(\omega_p = .4~\text{rad/s}\). - **Model:** We want to use a smaller rotor, but the model's angular speed \(\omega_m\) **must not exceed 50 rad/s**. What is the **minimum rotor radius \(R_m\) for the model**? - **Non-dimensionalize the power relation.** - **If the model generates \(P\) watts, what is the power of the prototype?** --- ## **Step 1: Non-dimensionalize the Power Relation** The **power generated** by a wind turbine is generally \[ P = f(\rho, V, R, \omega) \] We use **Buckingham π theorem**: **Dimensions:** - \([\rho] = M L^{-3}\) - \([V] = L T^{-1}\) - \([R] = L\) - \([\omega] = T^{-1}\) - \([P] = M L^2 T^{-3}\) We want to express \(P\) in non-dimensional form. Let’s construct dimensionless groups: Let’s guess a group: \(\frac{P}{\rho R^2 V^3}\) Check: - \([\rho R^2 V^3] = (M L^{-3})(L^2)(L^3 T^{-3}) = M L^2 T^{-3}\) - Same as [P]. So, \[ \pi_1 = \frac{P}{\rho R^2 V^3} \] The other relevant dimensionless group is the **tip speed ratio**: \[ \lambda = \frac{\omega R}{V} \] So, the non-dimensionalized relation is: \[ \frac{P}{\rho R^2 V^3} = F\left(\frac{\omega R}{V}\right) \] --- ## **Step 2: Model-Prototype Similarity (Matching \(\lambda\))** For **dynamic similarity**, the **tip speed ratio** must be the same for both model and prototype: \[ \lambda_p = \lambda_m \implies \frac{\omega_p R_p}{V_p} = \frac{\omega_m R_m}{V_m} \] **Given:** - \(R_p = 50~\text{m}\) - \(V_p = 5~\text{m/s}\) - \(\omega_p = .4~\text{rad/s}\) - \(V_m = 5~\text{m/s}\) (since “we do not want to perform the experiment at a wind speed other than 5 m/s”) **Unknown:** \(R_m\) (minimum, \(\omega_m \leq 50~\text{rad/s}\)) --- ## **Step 3: Find Minimum Model Rotor Radius** From similarity, \[ \frac{\omega_m R_m}{V_m} = \frac{\omega_p R_p}{V_p} \] Plug in known values: \[ \frac{\omega_m R_m}{5} = \frac{.4 \times 50}{5} = 4 \] \[ \omega_m R_m = 20 \] But \(\omega_m \leq 50\): \[ R_m = \frac{20}{\omega_m} \] to **minimize \(R_m\)**, use **maximum allowable \(\omega_m = 50\) rad/s**: \[ R_m = \frac{20}{50} = .4~\text{m} \] --- ## **Step 4: Scaling the Power** From the non-dimensional relation: \[ \frac{P_p}{\rho R_p^2 V_p^3} = \frac{P_m}{\rho R_m^2 V_m^3} \] \[ P_p = P_m \cdot \frac{R_p^2 V_p^3}{R_m^2 V_m^3} \] --- ## **Step 5: Final Answers** ### **A. Minimum Rotor Radius for the Model** \[ \boxed{R_m = .4~\text{m}} \] --- ### **B. Power Scaling** If model produces \(P_m = P\) watts, then the full-scale turbine would produce: \[ P_p = P \cdot \frac{R_p^2 V_p^3}{R_m^2 V_m^3} \] Plug in values: - \(R_p = 50~\text{m}\) - \(V_p = 5~\text{m/s}\) - \(R_m = .4~\text{m}\) - \(V_m = 5~\text{m/s}\) \[ P_p = P \cdot \frac{(50)^2 \cdot (5)^3}{(.4)^2 \cdot (5)^3} \] \[ P_p = P \cdot \frac{250}{.16} \] \[ P_p = P \cdot 15625 \] --- ## **Summary Table** | Quantity | Value/Expression | |----------|------------------| | Minimum \(R_m\) | \(.4~\text{m}\) | | Power scaling | \(P_p = 15625 \times P\) | --- ## **Final Answers** ### **1. Minimum rotor radius of the model:** \[ \boxed{.4~\text{m}} \] ### **2. Power scaling:** If the model generates \(P\) watts, the full-scale turbine generates: \[ \boxed{P_p = 15{,}625~P} \] Let me know if you need further explanation on any step!