Question

1.) When small particles of diameter d are transported by a moving fluid having a velocity V, they settle to the ground at some distance ( after starting from a height h as shown in the figure below. The variation 1 in ( with various factors is to be studied with a model having a length scale of Assume that 10 (= f(h,d,V,V, 4) where y is the particle specific weight and ļ is the fluid viscosity. The same fluid is to be used in both the model and the prototype, but y(model)= 9xy (prototipe). (a) If V = 45 mph, at what velocity should the model tests be run? (b) During a certain model test it was found that t(model) = 0.9 ft. What would be the predicted ( for this test? Particle (= f(h, d,V,7,1) CEL h=L d=L V=LT-1 FI-3 W=FC-21

Answer 1

Solution: given' t = f(h., d, V, V, 4) s we have total parameters n = 6 fundament al variables (m)=3 (f. 6, I] So number of repeating variables: 3 nama n-ms 63 = selecting u h, w as repeating variables: @ Ti, Ist in team; lusing "i"); elinio (visbi (u za (L'IPCPT="3!"(FE_75] Comparing powers on Team Lovo (2)'[(L') ( L both sides » f's power in C, so

ک'T د ,4+ ,- = . :Power و تا و 1- ده د معها 3-4 ,4+ :po en 1 - using a, b, c, in ego 1 : ( ) اح می را 71 : eln 2) Ind in term ( using "d") ar [ ریس درس راه = 7 آرامش در را) - 70•°F an د similarly" Comparing, مته = - =4 H2 = /h Tyed in term lusing -:("لا

Th3 = (ngos (1999 " [ khi Fºcºto Fil (L')}LL!=-)53[fc2j comparing powers on both sidesi f's power: 3. I +6=0 1 + co Y = -1 - T's power! =) - bz + 95o = bz=u → L's power, -3 +9z+big -29= =) - 3+93 -1 +2 50 = 93= 2 using a, b, sin ega 7y v the 'w' T3 у h So VM dength scale -nie bata given) ko fluid is same for both.. Mmap em. le

Vp = 45 mph I'm = 9vp , for similarity em lp Ке Tilm = () 5 hm ho حه hm also! (153) m = (113) 2 2 =) ym. hm Vp. he Vmodim Vp. Hp gye: Ypo tej um 145) a> m = 4.05 mph (6) given en = .9 ft lm 5 Yo = 2 fp = loxig (p = 9 ft