Inversin R. Allen Micro-hydropower Sourcebook

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log gate permits the forebay to be drained so that sedi-

ment can be cleared or repairs made. Because the

trashrack is located below the surface of the water,

much of the surface debris is not caught. The spillway

is strategically placed so that excess, overflowing water

continuously sweeps floating debris out of the forebay.

Less frequent cleaning of the trashrack is therefore

necessary. Just before the trashrack, stoplogs can be

inserted to prevent flow into the penstock when this is

necessary.

The design shows a vent incorporated in the

wall of the forebay; however, this vent is frequently

installed at the upper end of the penstock pipe itself

(see Fig. 5.21).

To reduce turbulence and losses, the penstock inlet of

large hydropower schemes is generally bell-shaped.

However, for many micro-hydropower plants, the pen-

stock inlet is simply a portion of the pipe extending

through, or flush with, the forebay wall. Even for these

configurations, the losses incurred are *usually minimal.

The head loss coefficient “K ” associated with various

pipe entrances is illustrated ?n Fig. 5.128. The actual

head loss is found by using the following equation:

he= K

e2g

(5.10)

where “v” is the velocity in the penstock. By incorpo-

rating a proper bell-shaped rather than a square-edged

entry, head losses at the inlet are significantly reduced.

But for penstock velocities even as high as 2 m/s, these

losses amount to only 0.1 m. Consequently, these losses

are generally insignificant, and only for very low-head

sites should an effort be made to incorporate a proper

bell-shaped entry. Even slightly rounding the entrance

reduces head loss by at least 50%. Although a hooded

inlet increases losses slightly, it makes it more difficult

for passersby to drop small stones into the penstock.

A vortex which forms at the inlet to the penstock occa-

sionally can cause troubles. It can induce loss of turbine

efficiency, possible cavitation, surging caused by the

formation and dissipation of vortices, and flow reduc-

tion as air replaces part of the water through the inlet.

It can also draw floating debris into the penstock.

Most research on the formation of vortices has involved

hydraulic model studies of specific sites. These are

appropriate for large hydropower schemes, but those

implementing micro-hydropower plants cannot afford

the cost and delays associated with such studies. Design

criteria to prevent vortex formation are needed, but

little has been done in this area.

Designing for a low velocity into the penstock and

increasing submergence of the inlet can help prevent

the formation of vortices. Also, a vertical inlet has a

greater tendency for vortex formation than a horizontal

one. Whereas these parameters are easy to quantify,

vortex formation also seems to depend considerably on

the circulation (swirl) in the water as it approaches the

inlet. This circulation is primarily a function of the

configuration of the area just upstream of the intake

and can be caused by canal irregularities or separation

of fiow at the edge of a canal or forebay wall. Flow

120 Civil works

approaching the intake asymmetrically is more prone to

vortex formation than symmetrical flow (Fig. 5.110). It

is therefore important that flows upstream of the inlet

area be as straight and uniform as possible.

Several studies have attempted to use empirical data

from both actual sites and model studies to derive

guidelines for minimum inlet submergence. Although

these efforts are not conclusive, they do indicate design

parameters for which vortices are less likely to form.

From data from 29 sites, one study derived the two

curves drawn in Fig. 5.111 (48). Regimes which are pre-

sumed free of vortices are found above each of these

curves. Also included in this figure are data points

obtained in a

recent study undertaken at the St.

Anthony Falls Hydraulic Laboratory (53). Based on this

more comprehensive study, the two previously derived

curves, which are still used for inlet design, are clearly

not adequate to specify vortex-free regimes. Had it

been possible to quantify approach conditions, more

accurate curves presumably could have been derived.

Although there is no area where vortices are certain not

to occur, the area bounded by the dashed lies indicates

an area where they are less likely to occur. Even in this

area, however, vortices may form if approach conditions

are poor.

Even if factors discouraging vortex formation are con-

sidered during the design stage, vortex flow can still

occur during operation of the plant. By th n, it may not

% be economically feasible to increase the su mergence of

the inlet, increase its diameter, or alter ap$roach condi-

tions to discourage vortex formation. In this case,

several remedial actions to suppress these vortices are

possible (94).

One of the simplest and least costly remedies is a float-

ing raft which disrupts the angular momentum of the

water near the surface (Fig. 5.112). It can be a square

grid built of strips of lumber. On one occasion, a float-

ing piece of plywood has also been used. However, if

this latter approach is taken, the piece of wood should

be large enough not to be sucked underwater and seal

the inlet to the penstock, especially if the penstock does

not include a vent pipe. In tests, a better location for a

raft was found to be slightly below the surface of the

water, Yrhere it is more effective in removing swirling

(a) 04

Fig.

5.110. Asymmetrical la)

and

symmetrical fb)

flow

approaching an intake.

3.a

2.c

0 horizontal intake

0 vertical intake

l n

intakes with problems

o 0 intakes with no problems

1.0

2.0

3.0

Fig. 5.111.

This

dimensionless

plot of data obtained from

intakes

existing installations and from

model

studies

proposed installations identifies

intakes

with and without

vortex problems (53).

motion below the raft. If the raft is submerged too

deeply, it will not be able to provide resistance to the

swirl above it.

A hooded inlet with baffle is another vortex suppressor

common to small intakes such as culverts. Figure 5.113

illustrates one such design used with culverts 150 mm or

greater in diameter (94). With the penstock velocities

commonly encountered in micro-hydropower installa-

tions, the pipe will then flow full even if submergence is

somewhat less than equal to pipe diameter, although the

(a)

Fig.

5.112. Section view of floating (a) and

submerged (b)

raft vortex suppressors.

Top:

baffle can be shaped

circular, square, or as shown

Side:

penstock pipe

Fzl

. .

co.75 D

Front:

all angle-iron braces

are40x40xbmm,\

Fig.

5.113. Three views of

a baffle on a hooded

Met usea

to Suppress vortex formation (9).

flow may still have some swirl. If a metal baffle is used

on a metal pipe, the same coating should be used on

each component to avoid cathodic corrosion.

Trashracks fabricated of bars of rectangular cross-sec-

tion can also disrupt the angular momentum of flow, and

suppress vortex formation. A variation of this is a

perforated plate (52). Laboratory tests have shown

that the head loss across a plate pierced with holes

through 50% of its area suppressed vortex formation

(Fig. 5.114). With low-head schemes where head losses

should be minimized, a perforated plate would not be an

appropriate method of vortex suppression.

Civil works 121

head loss

Lock

Ffg. 5.114. A perfomted plate

can also be used to suppress

vortex

fonnatfon

In Fig. 5.116, a simple design for a small scheme incor-

porates the basic features required

a forebay. In lieu

of a gate valve, a loosely fitting PVC (polyvinyl chlo-

ride) elbow is inserted between the penstock inlet and

the main penstock pipe. Flow to the turbine is cut off

Although micro-hydropower schemes generally require a

flow of less than 1.0 m3/s, the design shown in

Fig. 5.115 is proposed as a standardized design for flows

from 0.1 to 10 m3/s. Accompanying the original figure

is a table which assigns values to the variables repre-

sented by letters. The magnitude of these values

depends on the design flow the forebay is to accommo-

date. This is an example of a more sophisticated and

costly design that is simply reduced in size to accom-

modate the smaller flows used by micro-hydropower

plants. The previous design (see Fig. 5.109) has the

same basic features but is simpler and less costly. The

principal differences are the use of screw hoists on the

two gates and more intricate concrete work with the

standardized design.

Section AA

Section BB

Section CC

Fig. 5.115. A

standardized

fore&y design (VOO).

122 Civil works

LYTLA cover

cord for raising

penstock inlet

above water level

spillway

drain *

“‘IF-

\ \\ canal

loose 90° elbow

Fig. 5.116.

forebay

for a small plant

Colombia (49).

simply by pivoting the inlet end of the penstock out of

the water. As shown, the drain is used primarily to

drain the forebay and, depending on its size and loca-

tion, may permit scouring out any sediment; otherwise,

this sediment will have to be shoveled out. In addition,

the forebay can be covered because it is small. This

prevents debris from falling in and, possibly more

important, reduces chances of tampering by curious on-

lookers. In this design, a length of perforated PVC pipe

has replaced the conventional trashrack. In this case, it

is necessary to ensure that the total area through which

the flow enters is large enough tc keep the pressure

drop across this inlet within acceptable limits, even if it

is partially obstructed (see Trashracks and skimmers,

p. 162).

Another proposed forebay design is illustrated in

Fig. 5.117. In this design, a pipe section serves as a

cylinder gate through which the forebay can be drained

and debris and sediment removed. (See the end of Slid-

mg gatea, p. 155, for a discussion of cylinder gates.)

The pipe section can also serve as a shaft spillway if it

has been adequately sized (see Spillways,

159). The

operation of the circular trashrack is described in

Traahracks and skimmera (p. 162). If this forebay is also

to serve as a primary settling basin, it would have to be

adequately sized (see Settling basins,

166).

At times, no true

forebay is included before the pen-

stock. The installations shown on Figs. 5.118 and 5.119

are both low-head sites and use considerable flow. The

trashrack is placed at the end of the power canal. In

the first case, excess flow passes over a sliding gate; in

the second, part of this flow passes over the trashrack,

,-----

-----

OID :

-- --_

’

-- 1

----

-- --_

‘,

‘L-_ dC

,‘I

shaft spillway/ handle

drain plug 1

,trashrack

Section AA

-. - _.- -.

- -

r-lg. $.11/.

Pian

ana section views of a forebay

design

pro-

posed

by Rupert Armstrong EWns which features

a cylin-

drical

tmshrack

Fig. 5.118. At this

site,

the top edge

the sliding

gate

serves

the spillway. Tmsh is mked

off

the tmshmck

and pushed

off

the edge

into

the spillway canal.

Civil works 123

Fig. 5.119. The basic design is similar to

that in

Fig. 5.118,

but the

tmshmck has been lowered so that it is

submerged in normal opemtion The gate is shown

its

open position because the plant was

not

yet opemtional.

carrying away some of the debris which might other-

wise become lodged in the trashrack. In this regard,

Fig. 5.119 is similar to the design recommended by Oss-

berger (see Fig. 5.109), except that the latter is better

designed--all excess flow passes over the trashrack.

At other times, the forebay can be a substantial struc-

ture used to store water for peak generation purposes

(Fig. 5.120). Because a forebay used for this function

can be large, concrete is used only for the area around

the intake to the penstock. If the soil is pervious, the

entire forebay mlust b

e !ined with conrrete or other

impervious material. Fig. 5.121 shows a view of another

scheme which incorporates a forebay with significant

storage capacity. A close-up of the forebay (Fig. 5.122)

shows that concrete is only used around the inlet to the

penstock (at the left). The remainder of the forebay has

been carefully excavated and lined with a 10 cm-thick

layer of clay to seal it (Fig. 5.123). Hand-placed riprap

protects this clay lining (Fig. 5.124).

Fig 5.120.

A forebay used to

store water

for

power

gener-

a tion

during peak demand periods.

Fig. 5.121. A view

the forebay (with

4000

capacity),

pet-stock, and powerhouse at

80 kW scheme to provide

power to Syangja, Nepal.

Penstocks

Materials

Steel is the most frequently used penstock material, but

various types of concrete are also used. Wood-stave

penstocks with steel reinforcing rings, used extensively

in large, old plants, were entirely satisfactory. How-

ever, wood-stave pipe is now rarely used, probably

because of increased familiarity with steel penstocks.

For small schemes, the low cost, low weight, and ease

of installation of plastic pipe has made plastic an

attractive material. For

very

small self-help schemes,

pipes have also been constructed of other materials,

such as oil drums (see Fig. 10.53) and ordinary timber

(see Fig. 10.54). Properties of materials used for pen-

stock pipes are shown in Table 5.5 and will be referred

to in subsequent sections.

In selecting the type of penstock pipe for a specific site,

the following factors should be considered:

e required operating pressure and diameter;

o method of coupling;

o weight and ease in handling and accessibility of site;

o local availability of pipe;

o maintenance requirements and expected life;

a nature of terrain to be traversed; and

effect on pipe of water quality, climate, soil, and

possible tampering.

Although polyvinyl chloride (PVC) pipe has become more

popular, a source of concern with this material is its

degradation from exposure to ultraviolet radiation, pri-

marily from sunlight. This exposure can occur when a

portion

the pipe is installed above ground or during

shipmeut or storage at the job site.

Degradation occurs on the exposed surface of PVC pipe

where the sun’s radiation breaks the chemical bonds

within the polymer chains. The structure of the plastic

is permanently altered in the affected area, which often

124 Civil works

Fig. 5.122. Close-up of the forebay

under

construction Concrete is used only

in the

vicinity of the inlet to the penstock

turns the surface a light yellow. But penetration depth

of this altered area rarely exceeds 0.003 mm. Degrada-

tion ceases when exposure ends.

Tests were performed across the United States to gage

the effect of ultraviolet radiation on PVC pipe. During

the two years that the pipe samples were placed out.,

doors with maximum exposure to sunlight, both tensile

strength (which is directly related to pressure rating)

ant1 modulus of elasticity (which is related to the pipe’s

ability to resist external loads) remained virtually

unchanged. Impact strength was somewhat reduced but

still within acceptable limits (7).

Because long-term trends may be less clear, pipe

installed above ground should be protected from sunlight

by an opaque shield of ally thickness. The pipe can be

painted, coated, or wrapped. Because some paints and

cements contain solvents which can be detrimental to

PVC, it is advisable to apply the proposed paint or coat-

ing to a sample length of pipe to ensure compatibility

and adhesion. Generally, water-based latex paints for-

mulated for outdoor use work wel!.

Sizing

penstock

pipes

When sizing a penstock pipe, two parameters must be

specified:

o its diameter, which should be selected to reduce

frictional and therefore energy losses within the

penstock to an acceptable level, and

o the thickness of its walls, which shoulcl be selected

to accommodate the pressures encountered during

plant operation.

Selecting pipe diameter. A more widely used form of

the power equation is not Eq. (4.2) but

P = 9.8 et Q H

(5.11)

where, in addition to the variables defined earlier,

= efficiency of the turbine (or turbo-generating

equipment if electricity is to be generated)

H = net head acting on the turbine (m)

= Hg - h(

Fig. 5.123. A layer of clay (seen stacked

on the right) is

being tamped on the

floor

the foreboy.

Fig. 5.124.

The

clay lining of the forebay is covered with

hand-placed ripmp.

Civil works 125

TABLE 5.5. Paopertfea of mateiiala used in the manufacture of penstock pipe

Youq&mcldlllus coefficient

of elastici linearcxpandcm

Material

Et&f/= 1

a (m./m Oc)

Steel 21.105 12*10-4

Polyvinyl chloride (PVC) 0.28.lo5 54.164

Polyethylene 0.02-0.08.105 140 “lo-6

Concrete 2.105 10.10-6

Asbestos cement 8.1*10+

Cast iron 8~10~ 10.10’6

Ductile iron 17.105 11-10-6

Ultimate

tende s

b&?fEtb

3500

280 *

60-90 *

1400

3500

Elzz 1

‘9

7900

1400

940

1800-2500

1600-2100

77.00

7300

* Hydrostatic design basis

As the value admitting water to the turbine is just

beginning to open, the net head acting on the turbine is

equal to the gross head across the penstock, the differ-

ence in elevation between the penstock inlet and the

turbine. But as soon as the water in the penstock starts

to descend, fluid friction and turbulence result in losses

in head “h ” which reduce the actual head acting on the

turbine. I!.. be magnitude of these losses is approximately

proportional to the square of the velocity of the water

in the pipe. Consequently, if a less costly, smail+a-

meter pipe is used to convey the same flow as a larger-

diameter pipe, the velocity and therefore the losses are

greater in the former. Greater losses decrease a site’s

net head and therefore its power potential. Thus,

selecting a penstock diameter is a trade-off between

cost and power losses: selecting as small a diameter as

possible to minimize cost and selecting as large a dia-

meter as necessary to minimize losses.

As discussed in Selecting wall thickness (p* 1331, the

pipe diameter also affects the required pressure rating

and therefore the wall thickness of the penstock. A

smaller-diameter pipe would be less costly, but the

resulting higher velocity would cause pressure surges,

such as those caused by rapid gate closure, to be higher.

A more costly pipe with a higher pressure rating might

therefore be required. In selecting pipe diameter, it is

best not to consider the effect of diameter on pipe

thickness until an initial choice of pipe diameter is

made based solely on keeping head losses within accept-

able limits. The choice can always be changed if it is

later found to im-ly unacceptable surge pressures.

Pressure or b

>sses are comprised of losses caused

by friction bcrween the water and the pipe “hf” and

losses caused by turbulence created by changes in velo-

city or direction of flow which occur at the inlet, bends,

valves, and enlargements or constrictions along a pipe

“h n,

Losses caused by turbulence are commody

erred to as minor losses and are small in the case of

micro-hydropower schemes with relatively long pen-

stocks. However, if a short penstock is used, these

126 Civil works

“minor

losses” may be greater than friction losses. The

total head loss is simply equal to the sum of these two

losses:

hl = hf + ht (5.12)

One means of approximating these losses is described

below. The equations used to estimate these losses are

based on tests in both laboratories and full-scale instal-

lations.

The graph in Fig. 5.125 can be used to determine losses

caused by friction in a penstock pipe. It is based on the

following equation, which is derived from Manning’s

equation:

= 6.3s3 (5.13)

where

= head loss due to friction (m)

= penstock length (ml

n = roughness coefficient (see Fig. 5.126)

v = mean velocity (m/s)

D = internal pipe diameter (m)

The use of this graph is illustrated in EXAMPLE 5.4.

A more convenient form of this equation

expresses

the

losses in terms of the flow through the pipe “Q” (m3/s):

hf n2 Q2

r= lo-

D5.3

(5.14)

The very strong dependence of friction losses on pipe

diameter is more clearly seen in this form of the equa-

tion. Just halving the pipe diameter for a given flow,

for example, increases friction losses by a factor of 401

The equation also shows that, for a given flow and pipe

diameter, losses are proportional to penstock length.

0.5

0.2

0.1

0.05

\LLY?j

llllllillllllllll!llllll!

* i i i

i i i .kl Xi ” !+‘!-‘! *’ *! ’ /

0.002

0.0011 rmml

11 11 11

Fig. 5.125. Graph used

to determine head losses

caused by

frictfo? within a

pipe.

Civil works 127

Plastic

Welded steel

Wood stave

Concrete

very smooth

made from steel forms

with rough farms

Galvanized iron

Corrugated metal

n (roughness coefficient)

Fig. 5.126. The roughness coefficients

for

Manning’s for-

mula used in

determining head loss

caused by

friction

in a

pipe

good

condition

conveying

water.

This is one reason that hydropower schemes are laid out

to minimize penstock length for a given gross head (see

DEVELOPING THE BASIC LAYOUT,

p. 49).

Whereas Manning’s equation is used here, other empiri-

cally derived formulas also can be used, such as those of

Darcy-Weisbach, Hazen-Williams, and Scobey. Informa-

tion on these can be found in texts on hydraulics. Mann-

ing’s formula is widely used and can be applied to con-

duits of any shape, including canals (see

Determining

canal

dimensions

and slope, p.

96).

Although it can be

used only for water at normal temperatures, this

restriction has no impact on its use for hydropower

applications.

The 4% loss in power caused by friction ltisses in

EXAMPLE 5.4 is usually acceptable. One factor which

determines acceptability is an economic one--whether

the extra cost involved in installing a larger penstock to

reduce the energy losses can be recouped by the addi-

tional revenue generated by selling this additional

energy. This might be the case if all excess energy can

be sold or, equivalently, if it replaces diesel fuel at an

isolated site. In the United States, small suppliers of

power are guaranteed a market in the local utility for

their excess power; however, at mast isalated micro-

hydropower installations in developing countries, few

arguments can be made for reducing such small power

losses. At these sites, short-term peaks are usually bet-

ter addressed by better load management-spreading the

load more uniformly

during

the day-than by increasing

the initial capital outlay for larger-diameter pipe.

Clearly, each situation must be considered on its own

merits. EXAMPLE 5.5 presents a simple analysis of

whether an increased diameter pipe is warranted in the

case of a grid-connected scheme.

A&e ‘that d PVC pipe’ with an intemaE diameter of

2~50

mm is to convey a flbw

'60

P/s for u dfstance of

500 m at a site with a ghxk

head cif,8O.b m.

What

will .@e:the head loss due

friction,

arid what pep

centage loss in power does

this

head IO= represent?

From

Fig.

5.126, V-C pip& ,+ found to have ,a rough- ‘.

ness.coefficient’ of II =‘O.OlO: %ollowing the’elaqhed

l&e in Fig. 5.125 from the floF ,which the pipe is Qo

accomu+ate on the upper leftaand scale, a head

losi3

is found to be 0.0060 m/m of -pipe. The actual

head loss is then

hf 5 0.006OL =3.0m

and the net head is

H=Hg-ll*=80.0-3.0=77;0~m.

if bnli friction losses are consjdered.

Because the poyer‘available from a turbine with a

given flow is proportional to net head, this loss of

head represents a loss of power through pipe friction

of 3/80 or about 4%.

EXAMPLE 5.5

Assume that

in the case presented In EXAMPLE 5.4,

the owner

the power schem-e can sell power

for

US$ O.OS/kWh to the

grid.

Would it be adwntageous

for

him’.!0 spend more

for a larger-diameter

pipa

(300 mm~bt $30/m) which. would

reduce

head losses to

aiitji4t I:0 in than to use the 250 mm pipe

originally

pbnrk?d(~t$20/m)?

,J&reasiag the diameter ,of the pipe would mean that

approximately an additional

., I” -.

P=.-lOeQbH, I

= 10,(0.7~)(0..~6~(2.0) =: 0.8 kW

will be generate& +ssu+i& a plant factor $ 0.9,

this represents an annual revenue qf

< ,. ,’

(0.8 -k~,@.&8ikO h)($O.OS/kWh) A’$? 10

.The, increa&d price of the pipe alone would amount

tp. ^’ “-._

: ‘,

. .

The owner will then have to decide whether an addi-

tional investment of at least $5000 is worth an annual

return of

$310.

..-

128 Civil works

A more rigorous approach to selecting the optimum

pipe diameter-that diameter resulting in the least cost

over the life of the plant--is to select several possible

pipe diameters. First, for each diameter, power and

annual energy losses are computed as illustrated in

EXAMPLE 5.5. Losses caused by turbulence (discussed

below) should also be included in this calculation. The

present value of this energy loss over the life of the

plant is then calculated and plotted for each diameter

(Fig. 5.127). Second, the cost of the pipe for each dia-

meter is computed and plotted. These computations

should include all factors which might significantly

influence its cost. For example, decreasing the pen-

stock diameter for a given flow may appear to reduce

the price of the pipe; however, decreasing the diameter

also increases the velocity and potential surge pres-

sures, which may mean that a thicker, more expensive

pipe is required. Once both curves are plotted, they are

added graphically. The optimum diameter is the one

associated with the minimum cost. If standard, com-

mercial pipe is to be used, the optimum diameter would

be that diameter closest to the theoretical optimum,

Commercial pipe in non-standard sizes is either not

available or may cost significantly more to fabricate.

minimum cost

t--

cost of penstock,

valves, installation,

optimum

diameter

Pipe diameter

Fig. 5.127.

Graphical

rnethod for finding optimum

pen-

stock diameter.

If the internal diameter of the pipe changes a!ong the

length of the penstoch, the approach to determining

head losses described above can be applied to each con-

stant-diameter section. Because head losses are addi-

tive, total head loss caused by friction is simply the sum

of friction losses in the individual sections of pipe.

When pipe has to be shipped a considerable distance,

using pipe of two different diameters may save on proj-

ect costs. For a plant in Kosrae in the Federated States

of Micronesia, shipping c.Jsts for PVC pipe were approx-

imately halved because

tbe

15” pipe sections could be

telescoped into the 18” pipe sections, reducing the bulk

to be shipped by half. Volume, not weight, determines

the cost

shipping the pipe in containers. When the

penstock was assembled, all the pipe sections of one

diameter were secured together and laid, followed by

those of the second diameter. A reducer was used to

couple the two lengths of different diameter.

Losses caused by turbulence are generally expressed as

a product of a loss coefficient and velocity head:

ht=Kg

(5.15)

where

= head loss due to turbulence (m)

= head loss coefficient (see below)

= velocity where the loss is taking place (or the

higher of the two velocities if a contraction or

expansion occurs).

The coefficients for losses at entrances, valves,

bends, and contractions along a pipe are presented in

Fig. 5.128. Different sources show considerably differ-

ent values

for

these coefficients, and they should be

used only to obtain approximations of expected losses.

The loss coefficient for a 90° bend with a radius equal

to one pipe diameter, for example, is given in various

sources as O.Ll

(261, 0.35 (321, 0.4 (471, 0.50 (731,

and

1.0 (40). Clearly, the losses derived using these coeffi-

cients should not be considered correct to more than a

single significant figure. For example, for a 90’ bend

with a radius of three pipe diameters and a velocity of

2.0 m/s, the loss found using Eq. (5.15) should properly

be written as 0.8 m and not 0.0816 m. Because of the

meaning

“significant figures,” the loss could actually

0.5

& 1.0 (see APPENDIX B; p. 267).

Using Eq. (5.15), losses at points along a pipeline where

turbulence occurs can be calculated and added together

to determine the total loss resulting from turbulence.

The graph in Fig 5.129 permits approximating these

losses along a length of pipe of constant diameter. The

total

loss

coefficient, KT,

is simply the sum of all the

individual loss coefficients along the pipe.

If a smaller-diameter pipe is used for one portion of the

penstock, Figs. 5.125 and 5,129 can be used for each

constant-diameter portion. In this case, the contraction

loss caused by the change in diameter between the two

pcrtions of pipe should be included only when deriving

losses for the smaller-diameter portion. This is illus-

trated by means of EXAMPLE 5.6.

When a penstock configuration and the flow to be con-

veyed are given, the head loss can be determined

using the graphs in Figs. 5.125 and 5.129. However, as

was mentioned previously and as can

seen from

EXAMPLE 5.6, friction losses are usually significantly

greater than turbulence losses. In this case, turbulence

losses can be neg!ected and Fig. 5.125 alone can also be

used to approximate head losses with a given flow and

penstock configuration (Fig. 5.130a).

when the pipe

diameter is unknown, as is more frequently the case, an

approximate value can be obtained from Fig. 5.125 if

flow and maximum acceptable head loss (perhaps 5%

of gross head) are known (Fig. 5.130b) or if flow and

Civil works 129