Question
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When small particles of diameter d are transported by a moving fluid having a velocity V, they settle to the ground at some distance l after starting from a height h as shown below. The variation in l with various factors is to be studied with a model having a length scale of 1/10. Assume that: l = (h, d, V, γ, μ) where γ is the particle specific weight and μ is the fluid viscosity. The same fluid is to be used in both the model and the prototype, but γ (model) = 9 x γ (prototype). (a) If V = 50 mph, at what velocity should the model tests be run? (b) During a certain model test it was found that l (model) = 0.8 ft. What would be the predicted l for this
test? [Answer in units of mph and ft]
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Step 1

Following is the given information:

When small particles of diameter d are transported by a moving fluid having a velocity V, they settle to the ground at some distance l after starting from a height h as shown below. The variation in l with various factors is to be studied with a model having a length scale of 1/10. Assume that: l = (h, d, V, γ, μ) where γ is the particle specific weight and μ is the fluid viscosity. The same fluid is to be used in both the model and the prototype, but γ (model) = 9 x γ (prototype).

To find:

(a) If V = 50 mph, at what velocity should the model tests be run?

(b) During a certain model test, it was found that l (model) = 0.8 ft. What would be the predicted l for this test? [Answer in units of mph and ft]

Following velocity profile has been given to us

Step 2

Now finding the dimensions of the given variables as follows: help_outlineImage TranscriptioncloseLength,1 L] Diameter, d = [L] Height,h=[L] Velocity,V = [LT Specific weight,y =[ML2T] Viscosity,u =ML'T] fullscreen
Step 3

Now for finding the relation ... help_outlineImage TranscriptioncloseVariable Dimension Variable Dimension Variable Dimension Variable Dimension L] 0 l 0 0 h h h L L] d d d 0 d 0 0 h h LT [T*] [T'] V V V __ h h ML2T2yh ML'T h MLT] MLT [T] 0 yh The final relation between the variables is as follows: d Vu l hyh h ( fullscreen

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### Fluid Mechanics 