Question

The drag force Fp on a smooth sphere falling in water depends on the sphere speed V, the sphere density P. the density p and dynamic viscosity of water, the sphere diameter Dand the gravitational acceleration g. Using dimensional analysis with p. V and D as repeating variables, determine suitable dimensionless groups to obtain a reneral relationship between the drag force and the other variables. If the same sphere were to fall through air, determine the ratio of the drag force in water to the drag force in air to achieve dynamic similarity between these two systems. (7 marks) The density and dynamie viscosity of water are 1000 kg/m' and 1.12x10 N-s/m, respectively.

2 (a) The drag force Fp on a smooth sphere falling in water depends on the sphere speed V, the sphere density Ps, the density p and dynamic viscosity u of water, the sphere diameter D and the gravitational acceleration g. Using dimensional analysis with p, V and D as repeating variables, determine suitable dimensionless groups to obtain a general relationship between the drag force and the other variables. (8 marks) If the same sphere were to fall through air, determine the ratio of the drag force in water to the drag force in air to achieve dynamic similarity between these two systems. (7 marks) The density and dynamic viscosity of water are 1000 kg/m' and 1.12×103 N-s/m², respectively. The density and dynamic viscosity of air are 1.23 kg/m and 1.79×10 N•s/m, respectively. A viscous fluid flows in a 0.15 m diameter pipe is to be laminar. A pitot tube is used to measure the velocity within the pipe. Determine the radial location of the pitot tube that will measure the average velocity in the pipe. (b) (5 marks) At a radial location of 0.018 m away from the pipe wall, the measured velocity is 1.0 m/s, as shown in Figure 3. What is the flow rate and centreline velocity? (5 marks) Pitot Tube 0.018m ////- Pipe Wall v=1.0m/s 0.15m Flow -רדידדדלש Figure 3

Answer 1

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Date It giver: Tr - A (3 Third T terme ; On equating; b3 = -), 93 = -1 93 byt G-1-6 giver ; fourth T term; = sub - On equating i bazo 94 = -2, aa-3 b4 + 4 +1=8 It givex, - thung rear relationship is 6 if Aphere in 12 PVD to a fv21² y 428 VD; VD & Sf; fo Q P 4 = 1 falling through air : - FD Hof I Fo Tw? Ja