3. Flow of fluid through long pipes have been studied extensively. The pressure drop Ap required to drive flow through the pipe is a function of the pipe diameter D, the pipe length L, the pipe wall roughness € (measured in units of length), the average velocity of the fluid through the pipe V, the viscosity of the fluid µ, and the density of the fluid p. Using the II theorem, obtain the non-dimensional parameters in the flow. Use p, µ and D as the repeated variables.

(c) Note that V will appear on both sides of the equation. This form of equation is meant to find Ap when the average velocity V is known. However, let's say we know Ap and we wish to find V. The above form is then not useful as V appears in multiple terms. Since,non-dimensional numbers can be multiplied or divided by any other non-dimensional group to obtain a different non-dimensional group, then use this idea to re-arrange the function so that V-based parameter is isolated on the left-hand side and V appears only on the left hand side. (Note that, in doing this, the total number of non-dimensional groups should not change, just their form will be different!)2 (d) For 0.005, use the following data to make a plot of your new function, with yourvelocity parameter as an ordinate. Use Matlab or any other plotting software you arefamiliar with. Since the numbers have a wide range, it will be useful to plot log(Y)versus log(X), where Y and X are your velocity-based and pressure-based parameters,respectively. Here Rep = pVD/µ is the Reynolds number based on the diameter of thepipe.

(e) Use your plot to determine V in m/s, for the following flow: D = 5 cm, € = 0.025 cm,L = 10 m, for water with density p = 998kg/m³ and µ = 0.001kg/m.s. The pressure drop is deltap= 110kPa.