CIVE 240001

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This question paper consists of 3 printed pages, plus formula sheet, each of which is identified by the Code Number CIVE 240001

Graph Paper Formula Sheet

© UNIVERSITY OF LEEDS

May / June 2007

Examination for the degree of

BEng/ MEng Civil Engineering

Fluid Mechanics 2.2

Time allowed: 3 hours

Attempt 5 questions

1. A pipeline of length 1000 m and diameter 150 mm connects two water supply reservoirs together. If the value of the frictional factor, f, for the pipeline is 0.004 then determine the flow in the pipeline when the head difference between the two reservoirs is 20 m. [10 marks]

If an increase in demand requires that the flow be increased by 40% then

determine what length of the same diameter pipe is required to satisfy this demand if it is connected as shown in Figure 1. Assume that the friction factor for the new pipe is the same as the existing one. [10 marks]

Figure 1

20 m

New pipeline

CIVE 240001

2 Turn over

2. It is planned that a reservoir spillway will discharge into a long natural channel

which can be considered approximately rectangular and 20 m wide. At the toe of the spillway the maximum flow velocity is estimated to be 8 m/s with a corresponding depth of 0.25 m. Determine what length of the natural channel will have to be lined with concrete so that scour does not occur i.e. what length of channel is subject to supercritical flow. The bed slope of both channels is 1 in 200 and the Chezy C for the concrete and natural channels are 30 and 20 respectively. Use one step of integration. [15 marks]

In addition to protection from erosion what other factors would you need to take

into account in designing the concrete channel? [5 marks] 3. A pump is required to supply water to a header tank through a 50 mm diameter

pipe which is 100 m long and has a Darcy f equal to 0.006. It is necessary to have a continuous record of flow in the pipe therefore an orifice gauge is included

which causes a head loss of g

vp

24

2

where vp is the velocity in the pipe. The static

lift between the pump wet well and the header tank is 40 m. The relationship between head, H, and flow, Q, for the pump is given by the following equation

27.6004.2745 QQH −+=

Q in m3/s and H in metres.

(i) under these conditions determine the flow in the pipeline [15 marks] (ii) if the maximum efficiency of the pump is achieved when the flow is

5 /s then how could the system be redesigned so that the pump would operate at this efficiency [5 marks]

4. A centrifugal pump is supplying a 150 mm diameter pipeline which is 200 m long

measured from the outlet flange of the pump. The impeller of the pump is 0.3 m in diameter and 0.025 m wide at its outlet with backward facing blades inclined at 30o to the tangent. When the impeller rotates at 1600 RPM the suction lift including pipe friction losses in the suction pipe is 3.5 m. The head loss in the delivery pipe is 10 m and the static lift measured from the pump centre line to the outlet of the pipeline is 30 m. If the hydraulic and overall efficiencies of the pump are 75% and 50% respectively then determine the following:-

(a) the flow of water through the pump [12 marks]

(b) the shaft power required to drive the pump [3 marks]

(c) by assuming that the pipe is flowing in the fully rough zone determine the

mean height of roughnesses of the delivery pipe [5 marks]

CIVE 240001

3 240001 FORMULA SHEET

5. Two reservoirs are connected together by a pipeline which is 200 mm in diameter and 2000 m long for which the friction factor, f, is constant and equal to 0.0045. The head difference between the reservoirs is 25 m and remains constant. Under these conditions determine the flow in the pipeline. Minor losses can be ignored, however the rejected kinetic energy should be taken into account. [5 marks]

In this case is it worth taking into account the rejected kinetic energy?

[2 marks] Halfway along the pipeline there is a small fire hydrant which requires a

minimum flow of 7.5 ℓ/s and minimum pressure of 10 kN/m2 above atmospheric. Determine whether or not the pipeline is able to satisfy these conditions and if so what will be the flow downstream of the hydrant when it is in use. Assume that the level of the hydrant is 15 m below top water level of the upstream reservoir. In this case all minor losses and rejected kinetic energy can be ignored. [8 marks]

Describe with examples what is meant by minor losses in pipelines. When is

it necessary to allow for them and when can they be ignored? [5 marks] 6. It is required to measure the flow continuously in a long irrigation canal. The

canal is 8 m wide and can sensibly be considered to have a rectangular cross section for which the Chezy C is 40 in SI units; the bed slope of the channel is 1 in 2,750. A Venturi meter with a throat width of 3 m and a bed hump height of 0.6 m has been chosen for this purpose. When the depth of flow immediately upstream of the meter is 1.3 m then determine the following:-

(a) the discharge in the channel [5 marks] (b) the depth of flow at the highest point of the bed hump [5 marks]

(c) the normal depth in the canal [10 marks]

Assume that the coefficient of discharge for the meter is 0.9. 7. Describe the structure of the boundary layer as it occurs at the entrance to a

circular pipe. Use your description to show how a pipe may be hydraulically smooth under one set of flow conditions and hydraulically rough under another set. [10 marks]

A new pipeline of 450 mm diameter is required to carry a minimum flow of

150 ℓ/s of water. The pipeline has been designed to have a specific lifespan; if, at the end of its life, the mean height of roughness is expected to be 3 mm, what will be the flow in the pipeline when it is newly laid? Assume that the mean height of roughness when the pipe is new is 0.2 mm and that the head difference between the inlet and the outlet remains constant throughout the life of the pipeline. The Darcy f can be estimated from Prandtl’s rough pipe law. [10 marks]

CIVE 240001

4 240001 FORMULA SHEET

CIVE240001 – Fluid Mechanics 2.2

Formula Sheet Pipe Flow

Darcy-Weisbach formula 5

22

324

dflQ

gdvlfh f ==

Prandtl’s rough pipe formula 48.3log0.4110 +⎟⎟

⎠

⎞⎜⎜⎝

⎛=

skR

f

Colebrook-White formula ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

dk

fRfs

e 7.3251.2log0.41

10

Pumps

Work done per unit weight of fluid ( )122 1

1 uVuVg ww −=

Hydraulic efficiency 22 uV

gH

wh =η

Specific speed 4

3H

QNNs =

Open channel hydraulics Chezy formula miCAQ =

Manning’s formula 611 m

nC =

The backwater function nFji

dxdy

−−

=1

Specific energy g

vyE2

2+=

Critical depth flumes ( ) 23

71.1 hEBCQ d −=

Specific force g

AvxAF2

+=

Conjugate depth equation 2

8121

21

2

1nF

yy

++−=

(for rectangular channels)