chapter 2 incompressible flow through pipes

7/29/2019 Chapter 2 Incompressible Flow Through Pipes

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Chapter 2:

Analysis of Steady Flow in Pipelines



Learning outcomes

• To apply energy equation in pipes

• To analyze of flow and piping systems

including pipe in series and pipes in parallel by

applying energy equations

• To analyze pipe systems consist numerous

pipes connected in a complex manner with

general entry and withdrawal points by using

Quantity Balance Method

2UiTMKS/ FCE/ BCBidaun/ ECW301



Content

• Introduction

• Incompressible flow through pipes

•

Flow through pipes in series• Flow through pipes in parallel

• Flow through branching pipes

• Pipe networks• Quantity Balance Method




Introduction

• This chapter concerned with the analysis of flow in

pipes and piping systems including pipe in series and

pipes in parallel.

• Dealing with flow in circular pipes flowing full understeady conditions and flowing under gravity.

• Analyses problems involving pipelines, e.g. flow

between reservoirs, water flow from a reservoir

discharging to atmosphere, from higher to lower

elevation reservoir or vice versa.




Introduction

• Normally uses energy and continuity equation

to solve problems.

• Method of solving:

– Number of equations needed is the same as the

number of unknowns to be solved

– Successive corrections until the continuity

equation is satisfied (trial & error)




Incompressible flow through pipes

• Incompressible : density is constant throughout

• For a circular cross-section conduit flowing

full:

– Head loss due to friction by Darcy Weisbach

– Pressure loss by Darcy Weisbach

gd

fLvh f

2

42

(2.1)

d

v fL p

2

4 2 (2.2)




Incompressible flow through pipes

• But

•

Therefore (2.1) & (2.2),

4

2d

Qv

5

2

03.3 d

flQh f

5

224.3

d

flQ p

(2.3)

(2.4)




Example 2.1(Douglas, 2006)

• Water discharges from reservoir A through a 100 mm

diameter pipe, 15 m long, which rises to the highest

point at B, 1.5 m above the free surface of the

reservoir and discharges direct to the atmosphere atC, 4 m below the free surface at A. The length of pipe

L1 from A to B is 5 m and the length of pipe L2 from

B to C is 10 m. Both the entrance and exit of the pipe

are sharp and the value of f is 0.08. Calculate, – The mean velocity of the water leaving the pipe at point C

– The pressure at point B




UiTMKS/ FCE/ BCBidaun/ ECW301 9

A

2 m

B

4 m

C

L1

L2



Flow through pipes in series

• Pipes in series are pipes of different diameters

are connected end-to-end to form pipeline to

form a pipeline.

• The total loss of energy, or pressure loss, over

the whole pipeline will be the sum of the

losses for each pipe together with any

separation losses that might occur at the junctions.




Example 2.2 (Douglas, 2006)

• Two reservoirs A and B have a difference level of 9 m

and are connected by a pipeline 200 mm in diameter

over the first part AC, which is 15 m long, and then

250 mm diameter for CB, the remaining 45 m length.The entrance to and exit from the pipes are sharp and

the change of section at C is sudden. The friction

coefficient f is 0.01 for both pipes. List the losses of

head that occur and calculate the system flow rate.




Apr 2010

• Two pipes are connected in series and is used to connecttwo reservoirs as shown in Figure Q2(b). Both pipes are 500m long and have diameters of 0.7 m and 0.5 m respectively.The discharge from reservoir A to reservoir B is 0.8 m3 /sand the elevation of water surface in reservoir A is 85 m.Using the appropriate equations and assuming sharp exitand entry as well as existence of sudden contraction at the junction, determine :

– velocity for each pipe

– head losses due to friction, separation and fittings

– elevation of water surface in reservoir B

(Cc = 0.674, μ = 1.14 x 10-3 Pa.s and k = 0.15 mm)




Incompressible flow through pipes in parallel

• When two reservoirs are connected by two ormore pipes in parallel, the fluid can flow from oneto the other by a number of alternative routes.

• Difference in head h available to produce flowwill be the same for each pipe.

• Thus pipe can be considered separately,independent of other parallel pipes.

• Steady flow energy equation can be applied forflow by each route.

• The total volume rate of flow will be the sum of volume rates of flow in each pipe.





• Two sharp-ended pipes of diameter d1 = 50 mm,and d2 = 100 mm, each of length L = 100 m, areconnected in parallel between two reservoirs

which have a difference of level h = 10 m, as infigure. If the Darcy coefficient f = 0.008 for eachpipe, calculate,

a. the rate of flow for each pipe,

b. diameter D of a single pipe 100 m long thatwould give the same flow if it was substituted forthe original two pipes.




Incompressible flow through branching pipes

• Branching pipes - consist of one or more pipesthat separate into two or more pipes

• The basic principles that must be satisfied are:

• Continuity - at any junction the total mass flowrate towards the junction must be equal the totalmass flow rate away from it

• There can be only one head value at any point

• The energy equation must be satisfied for eachpipe




•

Figure 2.4 shows a simple branching pipesystem where there are three tanks connected

by three pipes that join at D

•

Actual direction of flow depends on – Tank pressures and elevation

– Diameters, length and kinds of pipe

•

The system must satisfy two equation i.e.Darcy Weisbach and continuity equation




•

The inflow and outflow at the junction D mustbe equal (continuity equation)

• The flow must be from the highest tank to the

lowest tank.

• Therefore, continuity equation can be

321321 or QQQQQQ




• If the HGL at junction D is above the elevationof the intermediate tank B – the water is

flowing into tank B.

• If the HGL at junction D is below tank B – thewater is flowing out from tank B.

• General problem: to find the flow rate in each

pipe




Example 2.4 (Douglas, 2006)•

Water flows from reservoir A through a pipe of diameter d1 = 120 mm and length L1 = 120 m to

junction D, from which a pipe of diameter d2 = 75

mm and length L2 = 60 m leads to reservoir B in

which the water level is 16 m below that in reservoirA. A third pipe of diameter d3 = 60 mm and length L3

= 40 m, leads from D to reservoir C, in which the

water level is 24 m below that in reservoir A. Taking f

= 0.01 for all the pipes and neglecting all losses otherthan those due to friction, determine the discharge in

each pipe.




Pipe networks

• A pipe network is a set

of pipes which are

interconnected so that

the flow from a giveninput or to a given outlet

may come through

several different routes.




• An alternative approach is used to analyse the

systems by means of successive approximations,

assuming values for the flow in each pipe, or theheads at each junction, and checking whether the

values chosen satisfy the requirements:

–

The loss of head between any two junctions mustbe the same for all routes between these junctions

– The inflow to each junction must equal the outflow

from that junction


Loss of

head in

pipe bc

+ Loss of

head in

pipe de

= Loss of

head in

pipe bd

+ Loss of

head in

pipe ce



Quantity balance method

• Can be used when the heads at various points in a

pipe network are known.

• Estimate the head at each junction and Q is calculated

for each pipe from the difference of head h betweenthe junctions at each end of pipe and its resistance

coefficient K by,

• If the inflow does not equal to outflow for each

junction, the original estimated head must be

corrected.

2KQh




• If the head at b has been overestimated by δh

relative to the head at a, c, and d, the values of

Qab and Qcb will be too small and the value of

Qbd will be too great.

• Differentiating


2KQh

hnh

QQ



• Therefore, the estimated head at b is reduced

by δh, the flows in ab and cb will be increased

to Qab + δQab and Qcb + δQcb.

• If the flows are correct, inflow and outflow at

b will be balanced:

bd bd cbcbabab QQQQQQ

bdinFlowcbinFlowabinFlow




• If hAD, hBD and hDC , are

the assumed losses in

pipe ab, cb and bd, used

to calculate Qab, Qcb andQbd,

hnh

QQ

ab

abab

hnhQQ

cb

cbcb

h

nh

QQ

bd

bd bd




• Substituting,

• Sign convention: flow towards junction b both Q and

h are posititve

nh

Q

Qh

nh

Q

nh

Q

nh

Q

QQQ

h

hnh

QQh

nh

QQh

nh

QQ

bd

bd

cb

cb

ab

ab

bd cd ab

bd

bd bd

cb

cbcb

ab

abab

bdinFlowcbinFlowabinFlow




• The process is repeated until ΣQ has been

reduced to negligible quantity (approaching

zero)

junctionthetowards

flowstheof sumAlgebraic DC BD AD QQQQ





• A reservoir A with its surface 60 m above the

datum supplies water to a junction D through a

300 mm diameter pipe, 1500 m long. From the

junction, a 250 mm diameter pipe, 800 m long,feeds reservoir B, in which the surface level is

30 m above datum, while a 200 mm diameter

pipe, 400 m long, feeds reservoir C, in which




Example 2.6 (Douglas, 2006)•

Water flows in the parallel pipe system as shown infigure for which the following data are available:


Pipe Diameter (m) Length (m) f

AaB 0.10 300 0.0060

AbB 0.15 250 0.0050

AcB 0.20 500 0.0050



• The supply pipe to point A is of 0.30 m

diameter and the mean velocity of water in it is

3 m/s. If the elevation of point A is 100 m and

the elevation of point B is 30 m above datum,calculate the pressure at point B if that at A is

200 kPa. What is the discharge in each pipe?

Neglect all minor losses.




Review of past semesters’

questions




Oct 2010

• Water flows from reservoir A through pipe 1 to

junction J which in turn flows to reservoir B

through pipe 2 and reservoir C through pipe 3

as shown in figure. Using analytical methodand neglecting all losses other than those due

to friction, determine the flow rate and velocity

for each pipe. Assume that the friction factor, f is 0.01 for all pipes.


chapter 2 incompressible flow through pipes

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