辅导案例-ENGR30002-Assignment 1

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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?
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