Fluid Mechanics ENGR30002 Assignment 1 Question One You start a business manufacturing paints. In a moment of clarity, you realise that you need only cyan, magenta and yellow to make all other colours. To this end, you install three tanks, one with each of these colours. You have also heard that a lot of mixing takes place in some pipe flows. Three tanks are connected to a common pipe. The individual pipes are made of stain- less steel (take the characteristic roughness as 0.03 mm). These are all connected by circular cross-section pipes with 2.5” internal diameters, to a common pipe of the same cross-section. The tanks have large diameters relative to these pipes. This is shown schematically in Figure 1. All tanks are open to the atmosphere. Figure 1: Schematic of paint production facility. Definitely not to scale. The flow rate of each paint is controlled by a throttling valve. A throttling valve can be thought of as a device of controllable loss coefficient. The valves are designed to introduce very little loss into the flow when they are fully open. 1 Table 1: System dimensions and fluid physical parameters. Variable Value Pipe internal diameter Lcm Lmy zt zc = zm = zy Lp Pipe roughness Paint density Paint viscosity 2.5 inch 3 m 3 m 1.5 m 3 m 25 m 0.03 mm 1200 kg m−3 0.1 Pa.s (a) Determine the flow rate of each paint, in litres per second, when its valve is fully open and the other two valves are closed. (b) What is the maximum rate at which you can make red paint? At this rate, what is the loss coefficient of each valve? Show all working. [Note: in subtractive colour mixing, as with paint, red is made by mixing equal parts magenta and yellow]. (c) Comment on the likelihood of delivering a well-mixed paint with this method. Describe, with reference to equations, ways in which the mixing could be improved in your system (without a separate mixing tank) while keeping the mass flow rate constant. Assume no minor losses associated with the merging of the pipes, the contraction from the tank into the pipe, nor with the 90 degree bends. Question Two (a) For a fixed mass flow rate of a given fluid flowing through a circular pipe, how does the specific frictional pressure loss (in J/kg) per unit pipe length vary with pipe diameter? Derive the relationship between frictional loss per unit length and the pipe diameter for both the laminar region and the fully-turbulent region with a fixed ratio of roughness (mm) to pipe diameter (mm). Use your result to qualitatively describe the relationship for fully turbulent pipe flow with fixed roughness (mm). (b) For a fixed volumetric flow rate through a fixed pipe diameter, how does the specific frictional loss (in J/kg) per unit pipe length vary with density? 2
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