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hw8_solutions

# hw8_solutions - ES330 Assignment 8 Solutions Chapter 7 Due...

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ES330 Assignment 8 Solutions Chapter 7 Due Date: Thursday October 28, 2010 II Problem 2 A student team is to design a human-powered submarine for a design competition. The overall length of the prototype submarine is L p , and its student designers hope that it can travel fully submerged through water at V p . The design team builds a one-tenth scale model to test in their university’s wind tunnel. A shield surrounds the drag balance strut so that the aerodynamic drag of the strut itself does not influence the measured drag. Write an expression for the air speed, V , they need to run the wind tunnel in order to achieve similarity. Solution To achieve similarity, the Reynolds numbers from the wind tunnel (model) and prototype must match. Therefore: Re m = Re p (1) ρ m V m L m μ m = ρ p V p L p μ p Solving for V m (or V ): V = V m = ρ p ρ m μ m μ p L p L m V p The length of the model tested in the wind tunnel is 1 / 10 the length of the prototype, so L m = (1 / 10) L p . Therefore, L p /L m = 10 in the above equation. V = 10 V p ρ p ρ m μ m μ p We could also further reduce the problem by using values of ρ and μ for air and water. Assuming that the air in the tunnel and the water used for the prototype are both at about 20 o C and atmospheric pressure, we can find the density and viscosity for both fluids. ρ air = ρ m = 1 . 204 kg/m 3 ρ water = ρ p = 998 kg/m 3 μ air = μ m = 1 . 825x10 - 5 kg/ ( m * s ) μ water = μ p = 1 . 002x10 - 3 kg/ ( m * s ) 1

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We can now solve for V : V = (10 V P ) 998 kg m 3 1 . 204 kg m 3 1 . 825x10 - 5 kg m * s 1 . 002x10 - 3 kg m * s V = 151 V p III Problem 3 The students from the previous problem measure the aerodynamic drag on their model submarine in the wind tunnel. They are careful to run the wind tunnel at conditions that ensure similarity with the prototype submarine. Their measured drag force is D m . Write an expression for the drag force on the prototype submarine from Problem 2. Solution To solve the problem, we can match: F d,m ρ m V 2 m L 2 m = F d,p ρ p V 2 p L 2 p (2) Solving for the drag force on the prototype, or F D,p : F D,p = ρ p ρ m L p L m 2 V p V m 2 F D,m We know V m and L p /L m = 10 from Problem 2, and F D,m is given as D m in the problem statement, so: F D,p = ρ p ρ m (10) 2 V p 10 V p ρ p ρ m μ m μ p !
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hw8_solutions - ES330 Assignment 8 Solutions Chapter 7 Due...

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