# Dynamic Similarity / Reynolds Number

1. Oct 24, 2009

### lizzyb

Question from Text

Gas (rho = 5.25 kg/m^3, v = 2.0 x 10^-5 m^2/s) is flowing in a 20-mm-diameter pipe. When a gas flowmeter measures the flow as being 0.064 kg/s, it registers a pressure drop of 8.5 kPa. Investigators plan to test an enlarged model that is geometrically similar in a 180-mm-diameter pipe.

(a) What flow rate at 25 'C water will achieve dynamic similarity?
(b) What would the pressure drop across the water meter be?

Relevant Equations/Info

When T=25'c, H2O: rho = 997.0 kg/m^3 ; v = 0.893 x 10^-6 m^2/s.

R = F_I/F_V

The book gives a table of "flow characteristics and similitude scale ratios (ratio of prototype quantity to model quantity)"; for answering (a), I'll use the one for Mass and Time:

Dimension M: (L^3 rho)_r
Dimension T: (L^2 rho / mu)_r

Work done thus far

So I'm wondering if I'm using the table correctly:

(M/T)_r = (L^3 rho)_r / (L^2 rho / mu)_r = (L mu)_r

expanding:

Flow_p / Flow_m = (L_p mu_p) / (L_m mu_m)

so

Flow_p = (L_p mu_p Flow_m) / (L_m mu_m) = (.18 * [997.0 * 0.893 x 10^-6] * 0.064) / (.02 * [ 5.25 * 2.0 x 10^-5 ]) = 4.88 kg/s.

I don't have an answer so I'm not sure if this is the correct. Does it look okay to you?
Would I do (b) similarly? That is, just use the table in like manner? Thank you.