ME2135/ME2135E
NATIONAL UNIVERSITY OF SINGAPORE
ME2135/ME2135E - FLUID MECHANICS II
(Semester II : AY2010/2011)
Time Allowed : 2.5 Hours
INSTRUCTIONS TO CANDIDATES:
1.
This examination paper contains FIVE (5) questions and comprises SEVEN (7) printed pages.
2.
Answer ALL FIVE (5) questions.
3.
All questions carry equal marks.
4.
materials: Students This is a CLOSED-BOOK EXAMINATION with authorized materials: are allowed to bring TWO (2) A4 size sheets of notes/formulae written on both sides.
5.
Programmable calculators are NOT allowed for this examination.
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QUESTION 1
The characteristics of a water pump, 0.5 m in diameter and operating at 750 rev/min, are experimentally generated, as given in Table below:
Q (m3/s) H (m) η (%)
0 15 0
0.006 16 30
0.012 16.5 55
0.018 16.5 70
0.024 15.5 76
0.030 13.5 70
0.036 10.5 57
0.042 7.0 38
0.052 0 0
(a)
This pump is to be used to supply water from one reservoir to another, the difference in water level being 8 m. The pipeline is 100 mm in diameter, 20 m in length, f = 0.002 and contains three flanged gate valves (k = 0.17) and one flanged check valve (k = 2.1). Determine the operating flow rate and the total head developed if the pump is operating at 850 rev/min.
(b)
The flow rate obtained in part (a) is subsequently deemed to be insufficient, even though the total head developed by the pump is adequate. You have a similar but smaller pump (0.45 m in diameter) at your disposable that can operate at 850 rev/min. Determine: (i)
How you should arrange the pumps in order to make best use of the pumps to boost the flow rate?
(ii)
The resultant flow rate in the flow delivery system.
(iii) The total power consumed by the pumps.
For graphical analysis, you may make use of the s ame graph paper that you have used for part (a). You may take density of water as 1000 kg/m 3 and standard acceleration of gravity g = 9.81 m/s2. (20 marks)
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QUESTION 2
Consider the flow inside a 90 degree corner. It has the stream function: r , 2r 2 sin 2 . (a)
Sketch a few streamlines.
(b)
Express the stream function in Cartesian coordinates.
(c)
The generalization of this expression is
r , Ar
,
sin
where A is an arbitrary constant and α is related to the geometry. What does this stream function represent? Sketch it. (d)
How does the speed in the corner depend on the α? (20 marks)
QUESTION 3
(a)
Consider a uniform flow U in the positive x-direction, with a point source of strength Q situated at (x,y) = (a,0) and a point sink of strength – Q at (-a,0). (i)
Find the location of any stagnation points in terms of Q, U and a.
(ii)
Does this location change with the relative strength of U and Q? If yes, discuss your result with the help of sketches of streamline patterns. (10 marks)
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QUESTION 3 (continue)
(b)
A vertical shaft carries a load of 320 kN on a thrust bearing consisting of 16 pivoted plates, 70 mm by 200 mm, arranged with their long dimension radially from the shaft and their plate centres on a circle of radius 0.30 m (similar to the sketch shown in Figure 1 below). The shaft rotates at 120 rpm and the dynamic viscosity of the lubricant is 0.1 Pa.s. If the plate takes the angle for maximum load (thrust) per unit width, neglecting the effects of path curvature and radial lubricant flow, find: (i)
the clearance between the pivoted plate and the fixed plate, and
(ii)
the total power loss in the bearing.
Note: For a given slider bearing of length L and inlet gap b 1, the maximum thrust per unit width is obtained when the inlet to exit clearance ratio b 1/b2 = 2.189, for which: F/w 0.1602
U L2 b
2 2
and D/w
0.754
U L b 2
All the symbols used have their usual meaning. (10 marks)
Figure 1
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QUESTION 4
A continuous planar belt of infinite width is used to skim or entrain a viscous liquid from a large container. The belt moves vertically upward with a constant velocity V as sketched in Figure 2 below, and drags up a film of liquid of thickness H, density ρ and dynamic viscosity µ. Gravity tends to make the liquid drain down the belt but the upward movement of the belt keeps the liquid from running off completely. The flow can be assumed to be laminar, incompressible, steady, parallel to the belt and fully developed with zero pressure gradient, and no shear is produced by the atmosphere at the outer surface of the liquid film. By using the co-ordinate system shown in the figure, and by applying the following equations: DV Dt
f
1 ρ
P ν 2 V
and
.V 0
where f is the body force vector per unit mass and other symbols have their usual meaning, express the velocity profile v in the liquid film (as sketched in the figure), the flow rate per unit width and the average velocity of the liquid film on one side of the belt only, in terms of V, µ, ρ, g and H. What is the condition for a net upward flow of liquid? (20 marks)
V
v y x Figure 2
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QUESTION 5
(a)
Explain briefly whether it is possible to have a laminar and a turbulent boundary layer at the same Reynolds number. If so, then: (i)
How do the velocity profiles compare? Sketch them on the same graph.
(ii)
Which boundary layer grows faster? Show it by considering boundary layer flows on a flat plate for Rex ( = ρUx/ µ) = 1x10 5 at x = 0.1m from the leading edge.
(iii)
Which boundary layer has the larger ratio of displacement thickness to momentum thickness (δ*/θ)? Provide the necessary information.
(iv)
When subjected to the same adverse pressure gradient, which boundary layer will separate sooner? Explain briefly.
Note: Assume turbulent boundary layer flow with one-seventh power law velocity profile. (7 marks)
(b)
From the Blasius solution of laminar boundary layer flow along a smooth flat plate, the skin friction (wall shear stress) τo on the plate at a distance x from its leading edge is given by:
τo 0.332 U
U
x
Using this expression of τ o, derive the expression for boundary layer momentum thickness growth rate θ/x in term of Re x ( = ρUx/ µ). All the symbols used have their usual meaning. (4 marks)
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QUESTION 5 (continue)
(c)
A thin sheet of fiberboard weighs 60 N and lies on a rooftop as shown in Figure 3. If the coefficient of solid friction between the board and the roof σ is 0.10, what wind velocity will generate sufficient friction force to dislodge the board? Assume turbulent boundary layer flow with one-seventh power law velocity profile, and check the validity of using this velocity profile assumption. For air, take the density ρ = 1.2 kg/m3 and the dynamic viscosity µ = 1.8 x 10-5 kg/ms. Note: For turbulent boundary layer flow with one-seventh power law velocity profile (5 x 105 ≤ Rex ≤ 107), the skin friction coefficient C f is given by:
Cf
o 1 2
ρU
2
0.0576 Re x
1/ 5
where all symbols have their usual meaning.
(9 marks)
U
Figure 3
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