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Find an expression for the gauge pressure at station 1 in terms of V₁, D₁, D2, d, e, and p. What assumptions do you have to make to solve this problem? What is the force, F, required to hold the cone in place? You may ignore the weight of the cone. Use the following values: V₁ = 20 m/s, D₁ = 30 cm, D₂ = 60 cm, and d = 58 cm. You may assume that the air density is at standard conditions, p = 1.225 kg/m³.

Find an expression for the gauge pressure at station 1 in terms of V₁, D₁, D2, d, e, and p. What assumptions do you have to make to solve this problem? What is the force, F, required to hold the cone in place? You may ignore the weight of the cone. Use the following values: V₁ = 20 m/s, D₁ = 30 cm, D₂ = 60 cm, and d = 58 cm. You may assume that the air density is at standard conditions, p = 1.225 kg/m³.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Problem 3: Air Flow in a Circular Duct with Conical Centerbody **Description:** Air is flowing down a circular duct as shown in the figure. At station 1, the velocity is \( V_1 \) and the duct diameter is \( D_1 \). The duct gradually enlarges to station 2, where the air exits to the atmosphere, and the diameter is \( D_2 \). A conical centerbody with a cone angle \( \theta = 30^\circ \) is inserted into the duct exit. The diameter of the cone base is \( d \). You may assume that the velocity is low enough that \( \rho \) (air density) is constant. **Figure Explanation:** - The diagram shows a circular duct with a control surface enclosing the air flow. - The air enters at station 1 with velocity \( V_1 \) and diameter \( D_1 \). - The duct then expands, and air exits at station 2 with velocity \( V_2 \) and diameter \( D_2 \). - The conical centerbody located at the exit has a cone angle \( \theta \) of \( 30^\circ \) and a base diameter \( d \). - The force \( F \) is acting horizontally to hold the cone in place. **Figure:** - The diagram visually represents the described scenario with a clear illustration of velocities, diameters at different positions, and the conical centerbody. **Given Data:** - \( V_1 \) = 20 m/s - \( D_1 \) = 30 cm - \( D_2 \) = 60 cm - \( d \) = 58 cm - Air density \( \rho \) = 1.225 kg/m³ (standard conditions) #### Questions: (a) **Find an expression for the gauge pressure at station 1 in terms of \( V_1, D_1, D_2, d, \theta, \) and \( \rho \). What assumptions do you have to make to solve this problem?** (b) **What is the force, \( F \), required to hold the cone in place? You may ignore the weight of the cone.** ---
Transcribed Image Text:### Problem 3: Air Flow in a Circular Duct with Conical Centerbody **Description:** Air is flowing down a circular duct as shown in the figure. At station 1, the velocity is \( V_1 \) and the duct diameter is \( D_1 \). The duct gradually enlarges to station 2, where the air exits to the atmosphere, and the diameter is \( D_2 \). A conical centerbody with a cone angle \( \theta = 30^\circ \) is inserted into the duct exit. The diameter of the cone base is \( d \). You may assume that the velocity is low enough that \( \rho \) (air density) is constant. **Figure Explanation:** - The diagram shows a circular duct with a control surface enclosing the air flow. - The air enters at station 1 with velocity \( V_1 \) and diameter \( D_1 \). - The duct then expands, and air exits at station 2 with velocity \( V_2 \) and diameter \( D_2 \). - The conical centerbody located at the exit has a cone angle \( \theta \) of \( 30^\circ \) and a base diameter \( d \). - The force \( F \) is acting horizontally to hold the cone in place. **Figure:** - The diagram visually represents the described scenario with a clear illustration of velocities, diameters at different positions, and the conical centerbody. **Given Data:** - \( V_1 \) = 20 m/s - \( D_1 \) = 30 cm - \( D_2 \) = 60 cm - \( d \) = 58 cm - Air density \( \rho \) = 1.225 kg/m³ (standard conditions) #### Questions: (a) **Find an expression for the gauge pressure at station 1 in terms of \( V_1, D_1, D_2, d, \theta, \) and \( \rho \). What assumptions do you have to make to solve this problem?** (b) **What is the force, \( F \), required to hold the cone in place? You may ignore the weight of the cone.** ---
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