A valve manufacturer has received an inquiry from a company seeking large units for an oil pipeline. Certain types currently in production might be suitable, but they are much smaller than what is needed. Before investing in expensive prototypes, the manufacturer wants to know what the flow resistances would be for enlarged versions of current designs. The measure of resistance is the pressure drop |ΔP| across an open valve (upstream minus downstream value). The manufacturer can readily measure |ΔP| as a function of the mean fluid velocity U in any existing unit, using either air or water at ambient conditions. The pipe diameter is D. You agree to provide guidance on extrapolating such data to valves for the oil pipeline.

(a) A good starting point is to assume that |ΔP| depends on D, U, ρ, μ, and, depending on the valve type, k internal dimensions, d₁, d₂,…, d_k. What can you conclude about |ΔP| from dimensional analysis?

(b) Suppose that for the oil pipeline D = 1.0 m, U = 2.0 m s⁻¹, ρ = 900 kg m ⁻³, and μ = 0.025 Pa · s. If the tests are done with 1⁄10-size versions of what is proposed, which is the better test fluid (air or water) and what would you recommend for U?

(c) A staff member points out that, for Re > 10⁵ (based on D), the company's experience is that |ΔP| is independent of μ. How would this alter your recommendations, if at all?

(d) If it were desired to predict |ΔP| for water in a pipe with D = 1.0 m and U = 2.0 m s⁻¹, how would your analysis and experimental recommendations change?