The drag, D, on a sphere located in a pipe flow through which a fluid is flowing to be determined experimentally. Assume that drag is a function of the sphere diameter, d, the pipe dianeter, H, the fluid velocity, V, and the fluid density, p.

Question

a) What dimensionless parameters would you use for this problem?
b)Same experiments using water indicate that for d=0.5cm H=1.27cm V=0.6m/s the drag is 7x10^-3 N. If possible estimate the drag on a sphere located in a 60cm diameter pipe through which water is flowing at V=1.8 m/s. Geometrical similarity is mainteined.

Answer 1

d/H had better be the same for model and experiment

If viscosity were important then rho vL/mu, the dimensionless Reynolds number should be the same, but the problem statement says mu is not important
so
we are left with d/H and the ratio of force to other parameters which we can derive as follows

force = f(d, p, v, h, )

d and H are not separate dimensions, ratio the same for similar geometries

= k * p^q v^r d^w

m l/s^2 = (m/l^3)^q * (l/s)^r * l^w

from mass, q = 1
ml/s^2 = m/l^3 *(1/s)^r *l^w

from time s
-2 = -r so r = 2

from length l
1 = -3+r+w
so w = 2

so in the end
Drag = k ( rho^1 v^2 length^2)

so I claim that if d/H is the same then
Drag/( rho v^2 H^2) will be the same for model and sphere
or
7*10^-3/[(10)^3 (.6)^2(1.27)^2]
= D/[ 10^3 (1.8)^2 (60)^2 ]

7*10^-3 = D (.36)(1.61)/[3.24*3600]

D = 140869 *10^-3
= 140.9 Newtons

Answer 2

By the way if gravity had been important (waves), we would have also found the Froude number v/sqrt(gL)

Answer 3

Thanks alot Damon :)

Answer 4

You are welcome :)