You are asked to find a set of dimensionless parameters to organize data from a laboratory experiment, in which a tank is drained through an orifice from initial liquid level h0. The time, τ , to drain a the tank depends on tank diameter, D, orifice diameter d, acceleration of gravity, g, liquid density, rho, and liquid viscosity, µ. How many dimensionless parameters will result? How many repeating variables must be selected to determine the dimensionless parameters? Obtain the Π parameter that contains the viscosity

Step 1

Given: A tank is drained through an orifice from initial liquid level (h₀). The time(r), to  drain the tank, depends on tank diameter (D), orifice diameter (d), acceleration due to gravity (g), liquid density (ρ) and liquid viscosity  (μ).

r = f (h₀, D, d, g, ρ, μ)

Step 2

There are totally seven parameters r, h₀, D, d, g, ρ and μ

∴ n = 7

The primary variables are M, L, T

∴ m = 3

So, the number of repeating variables are, r = 3

Select the three repeating variables are ρ, d, g

Then according to Buckingham, there will be n-m (⇒ 7-3 = 4) dimensionless groups given by π₁, π₂, π₃ and π₄

So, The number of π terms = 4

Explanation:

See attached images for step 3 to 7

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