# We conclude that there should be a relationship of

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We conclude that there should be a relationship of the form Δ P ρ U 2 = f Re, Π 1 , Π 2 ,... Π k ( ) or
Δ P = ρ U 2 f Re, Π 1 , Π 2 ,... Π k ( ) . As long as Re and the P i groups in this relationship are the same, the pressure drop measured in the tests should be the same as the pressure drop in the actual pipeline. b) In the actual pipeline, Re = ρ UD μ = 900kg.m 3 ( ) 2m.s 1 ( ) 1m ( ) 0.025kg.m 1 s 1 = 7.2x10 4 . In the tests, the dimensions will all be scaled down by a factor of 10, so the dimensionless groups P I corresponding to length ratios will be identical. We also need the Reynolds number to be the same in the tests as in the actual pipeline. Using the material properties given in Tables 1.1 and 1.2, for air we have U(air) = Re ν a D = 7.2x10 4 1.58x10 5 m 2 s 1 ( ) 0.1m = 11.4m / s and for water we have U(water) = Re ν w D = 7.2x10 4 1.00x10 6 m 2 s 1 ( ) 0.1m = 0.72m / s . Both of these velocities are reasonable. The speed of sound in air is about 343 m/s at room temperature, and so long as air flows at significantly lower velocities than that (or the Mach number , the ratio of the fluid velocity to the speed of sound in that fluid, is small, Ma << 1, see p. 15), air flows like an incompressible fluid. Still, the lower velocity of the water and the fact that, like oil, it is a liquid, probably make it a better choice. c) Since the Reynolds number here is less than 100,000, this observation does not affect the recommendation here. For Re>100,000, this observation indicates that Re is unimportant, so the tests can be done at whatever Re is most convenient—the lowest possible value, or just slightly above 100,000. d) Now suppose the pipeline were carrying water instead of oil, but we still wish to develop the valve by doing tests at 1/10 th the actual length scale. For water we have Re = UD ν w = 2x10 6 (water) .
This value of Re indicates we do not have to keep Re constant in doing the tests. If we did still wish to keep Re constant, we would have to use the velocities U(air) = Re ν a D = 2x10 6 1.58x10 5 m 2 s 1 ( ) 0.1m = 316m / s and U(water) = Re ν w D = 2x10 6 1.00x10 6 m 2 s 1 ( ) 0.1m = 20m / s . This velocity for air is comparable to the speed of sound, and would make air a poor choice as a test fluid. The best recommendation in this case would be to use water as the test fluid, and conduct the tests at a Reynolds number just above 100,000.