QUESTION 4
NATIONAL UNIVERSITY OF SINGAPORE
EXAMINATION FOR THE DEGREE OF B.ENG. (Semester I: 2011-2012)
CE2134 Hydraulics
Nov/ Dec 2011 - Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES 1.
This examination paper contains FIVE(5) questions and comprises SIX(6) printed pages.
2.
Answer ALL questions.
3.
All questions carry equal marks.
4.
This is an “OPEN BOOK” examination.
.../2
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CE2134
Question 1
Fig. Q1
A 0.15 m diameter pipe transports petroleum products from a refinery to storage tanks. The roughness of the pipe is estimated as ks = 0.12 mm (or e = 0.12 mm as in some texts). The steady flow rate is 0.045 m3/s. (a)
When the pipe carries only gasoline with a specific gravity of 0.68 and kinematic viscosity of 0.56 mm2/s, what is the pressure drop for every 100 m length of pipe? [10 marks]
(b)
When carrying petroleum products, gasoline is known to follow kerosene which has a specific gravity of 0.81 and a kinematic viscosity of 2.8 mm2/s with a fairly well defined and sharp interface as shown in the Fig. Q1. At a particular instant of time, the interface is located between two pressure gauges located at A and B which are 1100 m apart. At that instant, the gauge pressures at A and B are respectively 450 kN/m2 and 100 kN/m2. Determine the location of the interface at the given instant. [15 marks]
Note: The pressure drop ∆p in a horizontal circular pipe of diameter D over a distance is 2 f V given by ∆p = ρgh where h = . V =average velocity of flow in the pipe, f =friction D 2g factor, ρ =mass density of the liquid carried by the pipe.
.../3
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CE2134
Question 2
Fig. Q2
A sprinkler has two arms of unequal length and it is connected to a tap where the inflow is Q m3/sec as shown in Fig. Q2. Each arm has a nozzle of diameter d = 5 mm . (a)
What is the inflow Q if the torque to hold the sprinkler stationary is 0.036 Nm? Assume that the flow in each arm is the same. [10 marks]
(c)
If the torque on the sprinkler is released, what is the angular velocity of rotation assuming no friction? [10 marks]
(c)
What is the frictional torque if the rotational speed is reduced by 20%?
[5 marks]
.../4
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CE2134
Question 3 Consider steady laminar flow of incompressible fluid between two concentric, rotating cylinders. The radius and angular velocity of the inner cylinder are R1 and Ω1 and those of the outer cylinder are R2 and Ω2.
r uθ R1
R2
Ω1 Ω2
(a)
Determine the velocity profile uθ(r) provided that the momentum equation in the θ direction is d 1 d µ (ruθ ) = 0 dr r dr where μ is the coefficient of viscosity. [15 marks]
(b)
The above momentum equation also applies to the steady laminar flow of an incompressible fluid inside one single rotating cylinder with a radius of R and an angular velocity of Ω. Determine the corresponding values of R1, Ω1, R2 and Ω2 and derive the velocity profile uθ(r). [10 marks]
…/5
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CE2134
Question 4 A smooth flat thin plate aligned parallel to a steady stream of laminar flow of incompressible an fluid where pressure is uniform. Boundary layers develop along both sides the plate and form friction drag. Um
(a) The boundary layer velocity profile is approximated as 3
u y y = α1 + α 2 + α 3 Um δ δ where Um is the mainstream velocity (m/s) and δ is the boundary layer thickness (m). Determine the values of the coefficients (i.e., alphas). [8 marks]
(b) What is the general expression for the surface shear stress for the smooth flat plate? [8 marks]
(c) Determine the total drag force on the plate and express the drag coefficient, CD, in terms of the Reynolds number. [9 marks]
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CE2134
Question 5 Consider groundwater flow in a confined aquifer having a uniform flow of U (m/s) at a large distance upstream. A well situated at (0, 0) extracts water at a rate of q (m2/s) and acts as a sink. Groundwater flows very slowly and behaves close to an ideal fluid. (a) What are the stream and the potential functions of the combined flow field? [5 marks] (b) The stagnation point is at point at which velocity is zero. In other words, the location at which the uniform flow and the flow induced by the well cancel out with each other. Determine the location of the stagnation point. [5 marks] (c) Identify the streamline that passes through the stagnation point, and that the boundary of 2πUy y . the capture zone is = tan x q [10 marks] (d) Express the width of the capture zone at a large distance upstream, b, in terms of U and q by considering the continuity of flow. [5 marks] y U
capture zone boundary
b
x
-q well stagnation point
- END OF PAPER -
EXAMINATION FOR THE DEGREE OF B.ENG. (Semester I: 2011-2012)
CE2134 Hydraulics
Nov/ Dec 2011 - Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES 1.
This examination paper contains FIVE(5) questions and comprises SIX(6) printed pages.
2.
Answer ALL questions.
3.
All questions carry equal marks.
4.
This is an “OPEN BOOK” examination.
.../2
-2-
CE2134
Question 1
Fig. Q1
A 0.15 m diameter pipe transports petroleum products from a refinery to storage tanks. The roughness of the pipe is estimated as ks = 0.12 mm (or e = 0.12 mm as in some texts). The steady flow rate is 0.045 m3/s. (a)
When the pipe carries only gasoline with a specific gravity of 0.68 and kinematic viscosity of 0.56 mm2/s, what is the pressure drop for every 100 m length of pipe? [10 marks]
(b)
When carrying petroleum products, gasoline is known to follow kerosene which has a specific gravity of 0.81 and a kinematic viscosity of 2.8 mm2/s with a fairly well defined and sharp interface as shown in the Fig. Q1. At a particular instant of time, the interface is located between two pressure gauges located at A and B which are 1100 m apart. At that instant, the gauge pressures at A and B are respectively 450 kN/m2 and 100 kN/m2. Determine the location of the interface at the given instant. [15 marks]
Note: The pressure drop ∆p in a horizontal circular pipe of diameter D over a distance is 2 f V given by ∆p = ρgh where h = . V =average velocity of flow in the pipe, f =friction D 2g factor, ρ =mass density of the liquid carried by the pipe.
.../3
-3-
CE2134
Question 2
Fig. Q2
A sprinkler has two arms of unequal length and it is connected to a tap where the inflow is Q m3/sec as shown in Fig. Q2. Each arm has a nozzle of diameter d = 5 mm . (a)
What is the inflow Q if the torque to hold the sprinkler stationary is 0.036 Nm? Assume that the flow in each arm is the same. [10 marks]
(c)
If the torque on the sprinkler is released, what is the angular velocity of rotation assuming no friction? [10 marks]
(c)
What is the frictional torque if the rotational speed is reduced by 20%?
[5 marks]
.../4
-4-
CE2134
Question 3 Consider steady laminar flow of incompressible fluid between two concentric, rotating cylinders. The radius and angular velocity of the inner cylinder are R1 and Ω1 and those of the outer cylinder are R2 and Ω2.
r uθ R1
R2
Ω1 Ω2
(a)
Determine the velocity profile uθ(r) provided that the momentum equation in the θ direction is d 1 d µ (ruθ ) = 0 dr r dr where μ is the coefficient of viscosity. [15 marks]
(b)
The above momentum equation also applies to the steady laminar flow of an incompressible fluid inside one single rotating cylinder with a radius of R and an angular velocity of Ω. Determine the corresponding values of R1, Ω1, R2 and Ω2 and derive the velocity profile uθ(r). [10 marks]
…/5
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CE2134
Question 4 A smooth flat thin plate aligned parallel to a steady stream of laminar flow of incompressible an fluid where pressure is uniform. Boundary layers develop along both sides the plate and form friction drag. Um
(a) The boundary layer velocity profile is approximated as 3
u y y = α1 + α 2 + α 3 Um δ δ where Um is the mainstream velocity (m/s) and δ is the boundary layer thickness (m). Determine the values of the coefficients (i.e., alphas). [8 marks]
(b) What is the general expression for the surface shear stress for the smooth flat plate? [8 marks]
(c) Determine the total drag force on the plate and express the drag coefficient, CD, in terms of the Reynolds number. [9 marks]
…/6
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CE2134
Question 5 Consider groundwater flow in a confined aquifer having a uniform flow of U (m/s) at a large distance upstream. A well situated at (0, 0) extracts water at a rate of q (m2/s) and acts as a sink. Groundwater flows very slowly and behaves close to an ideal fluid. (a) What are the stream and the potential functions of the combined flow field? [5 marks] (b) The stagnation point is at point at which velocity is zero. In other words, the location at which the uniform flow and the flow induced by the well cancel out with each other. Determine the location of the stagnation point. [5 marks] (c) Identify the streamline that passes through the stagnation point, and that the boundary of 2πUy y . the capture zone is = tan x q [10 marks] (d) Express the width of the capture zone at a large distance upstream, b, in terms of U and q by considering the continuity of flow. [5 marks] y U
capture zone boundary
b
x
-q well stagnation point
- END OF PAPER -
