Warnick C.C. Hydropower engineering

Подождите немного. Документ загружается.

Water Passages

Chap.

of 9000 kW, a full-gate discharge of

1970

ft3/sec

(55.8

m3/sec), and arated net

head of 60 ft (1 8.3

m).

Indicate the approximate required space for accom-

modating the turbines.

8.3. Prepare a preliminary plan drawing for the required spiral case in Problem 8.2.

8.4. From a hydropower plant in your vicinity, obtain data to determine the

various controlling dimensions of the spiral case and draft tube similar to

those listed in Figs.

8.9

through 8.13 and check at least two dimension values

with the empirical equations presented for spiral case and draft tube size

determination.

8.5.

Prepare a neat line drawing for generalized dimensioning of draft tubes for

bulb turbine unit or tubular turbine unit and prepare a flow diagram of how

you would develop the empirical equations and diagrams similar to the work

deSiervo and deleva.

8.6.

very small hydropower site has a rated potential capacity of

250

kW and is

to have two penstocks.

the length of the penstock is 3700 ft

128

rn)

and

the gross head is 405 ft (123.4

m),

select suitable penstocks assuming the

turbine efficiency will be 85%. Make all necessary assumptions to determine

suitable penstock.

SYNCHRONOUS GENERATORS

Converting water energy to electric energy at hydropower plants is possible through

the operation and functioning of electrical generators. The phenomenon of produc-

ing

electrical current

a conductor, discovered by Michael Faraday, involves

moving a copper coil through a stationary magnetic field or moving a magnet

through a copper coil. In the practical generator, an induced voltage is caused by

the magnetic field of a rotor sweeping by the coils of the stator. The rotor of an

electrical generator in the case of hydropower developments is driven by the rota-

tion of the turbine. Usually, the turbine and generator are directly connected on a

common shaft. Most generators used in hydropower developments are

alternating-

current (ac) synchronous generators. These require excitation current which is

usually provided by a small auxiliary generator that supplies direct current to create

the magnetic field of the rotor.

Generator Components

The basic components of generators as used with hydraulic turbines are (1)

supporting frame,

(2)

shaft that transmits the rotating motion of the turbine,

(3)

exciter,

(4)

assembly of the built-up rotor,

(5)

rotor poles,

(6)

collector rings and

brushes or solid-state rectifiers,

(7)

stator and

its

component

coils,

(8)

stator coil

supports,

(9)

air cooler, (10) tllrust bearings for vertical shaft machines, and (1 1)

brake. These various components are indicated in the cutaway diagrammatic draw-

ing of

Fig.

9.1.

Elementary Electrical Considerations Chap.

Synchronous Generators

Collector ring and brushes

;

Statw coil support

Bearing

Shaft

Figure

9.1

Cutaway diagrammatic drawing of synchronous electrical generator.

The stator contains the armature in conventio~ial ac generators and consists of

windings of coils pressed into slots in a

syrn~iietrical pattern. The core of the stator

is composed of laminated steel sheets to reduce power losses by hysteresis and eddy

currents.

The rotor contains the coils that make up the electromagnets or field winding.

The windings surround the individual poles that are mounted on a structure that

makes up a wheel attached to the rotary shaft. If the windings surround each pole

in a symmetrical fashion and are wound individually around a pole that extends out

from a cylindrical surface, they are termed

soliet~t-pole

fields.

The number of coils, the wire size and number of turns in a coil, and the

number of slots in a stator are design considerations by whjch size and capacity

of the generator are varied. Sonietimes a double layer of windings is pressed into

the slots. These windings

may be connected in series

01.

parallel to achieve the

desired voltage or current ratings. For actual

functio~ijng of the generator, it is then

necessary to have twice ttie

number of coiis per phase per pair of poles.

The

phase

refers to the manner in which tile electrical currentis taken from

the windings.

single-phase current is a situation or arrangement in which electric

current

1s taken from the generator with one armature winding and delivers electric

current to two wires.

two-phase generator has two annature windings generating

two single phases of current

electrical degrees apart in phase, and the output

electric current is connected to a four-wire system.

three-phase generator has

three

amiature wilidings, delivering single-phase alternating currents,

120

electrical

degrees apart

phase, to a three-wire system. Different kinds of connections are

Delta connection Wve connection

Armature or stator

(The connection shown on this diagram are

wye connections with

OA,

OB,

and

common)

Figure

9.2

Schematic diagram of a four-pole

generator with connection

diagrams.

used to connect the windings.

delta,

connection has each winding connected

to the line wire and also connected to the next winding.

wye,

connection has

one end of each winding connected to similar ends of the other windings with

the

other ends connected to the line wires. The common connection may be used as a

grounding point, or as a neutral for a four-wire system. Figure

9.2

shows a sche-

matic diagram of a simple four-pole generator with line diagrams showing the

two

types of connections.

To illustrate the functioning of the generator, consider phase

windings

starting at location

OA.

The symbol x denotes that the winding is going downward

through the armature. The symbol

indicates that the coil winding is coming up-

ward through a slot in the armature. Referring to Fig.

9.2,

the phase

coil con-

ductor starts at point

and passes downward through a slot in the ar~nature next

to rotor pole

goes across the bottom of the ani~ature (lines shown dotted), and

comes up in a slot next to rotor

pole

N2,

then goes across the top of the armature

(lines shown solid) to a slot next to rotor pole

Sf,

and then downward. This con-

152

Elementary Electrical Considerations Chap. 9

tinues until the coil winding ends up back at the next slot marked

OA.

The slot

spacing and number of windings are design variables for changing the output of the

generator. The

and

phases take a similar relative path.

The field winding

magnetic circuit and rotor consists of poles that are dupli-

cates of each other except that they are arranged alternately north and south

magnetically.

full cycle of alternating current is developed for each pair of

magnetic poles swept by the winding, that is, one cycle per two poles. The fixed

number of poles is provided in a full circle and must be an even-integer

number of

poles, because a north pole must exist for each south pole. The following funda-

mental formula must be met:

where

frequency,

cycles/sec)

number of poles

speed, rpm

or,

irrotative speed,

is in radians per second,

There are only a limited number of frequencies used for ac power frequencies. The

usual ones are

25, 50, 60,

and

400

Hz. The most common frequency used with

hydropower generators in North America is

Hz.

Power

Factor

In most operations of generators it is important to recognize the concept of

power factor.

usual generator circuits there is inductive reactance and the load

current lags or leads the voltage by an angle,

measured as a part of the

360'

electrical cycle. The value of cos

is the power factor,

PF.

To illustrate this, con-

sider the

simple ac diagram in Fig.

9.3.

To understand the effect of power factor on synchronous ac generators, it is

necessary to consider three conditions:

(1)

unity power factor load,

(2)

lagging

power factor load, and

(3)

leading power factor load. Additional knowledge of the

characteristics of ac generators is needed to explain fully the importance of these

three conditions of power factor. The strength of the magnetic field varies directly

with the current in amperes passing through the rotor winding, and generator

voltage output will depend on the amperes passing through the rotor coils. Figure

9.4

illustrates how generator voltage for a

20,000-kVA

generator varies with the

rotor current at different power factors. If the rotor current of the generator is

considered to be constant at

270

while the load increases from no-load to full-

load, the voltage at the generator terminals will decrease. Referring to Fig.

9.4,

it is

noted

that

the decrease

from

13,200

at point

9000 V

at point

for

Synchronous Generators

Figure

9.3

current diagram showing alternating

sign

and electrical

lag.

the unity, or

100%

power factor. This change in voltage is called the

regularior~

the generator. This is usually expressed as a percentage:

3,200

9,400)

100

Change in voltage

28.8%

13,200

This voltage drop in the armature coils caused by resistance and reactance gives the

impedance drop

of the armature at

the particular load being carried.

Tile

im-

pedance drop for an armature can be expressed mathematically by the following

formula:

~(IR)~

(1x1~

(9.3)

No load

Full load

100% P.F.

Full load

90% P.F. lagging

Full load

0% P.F. lagging

Field Amperes

Figure

9.4

Cl~aractcristic curves for 20,000

generator.

SOURCE:

U.S.

Bureau of Reclamation.

154

Elementary Electrical Considerations Chap.

where

impedance drop,

armature current,

armature impedance, S2

armature effective resistance,

armature reactance,.n.

The armature effective resistance drop is in phase with the armature current. Induc-

tive

rcactance drop is at riglit angles, or 90°, out of phase with (and leading) the

armature current. Figure 9.5 shows how the three conditions of power factor load

la) Generator with unity power factor load

(b)

Generator with lagging

power factor load

Icl Generator with leadins

power factor load

Terminal voltage

Generator voltage

Armature current

Effective resistance drop

Reactance drop

Impedance drop Figure

9.5

Vector diagrams showing

Power factor angle

generator

voltagc variation with load

Power angle conditions.

Synchronous Generators

155

variation cause changes in the generated voltage. The magnitude of reactance and.

resistance drop depend on design factors such as slot depth, conductor size, spacing

of conductors, and the type of coils.

Additional understanding of generator performance. generator losses, and

power factor can be obtained by referring to Fig. 9.6. Here it is noted that the

power of the generator output is computed by multiplying the voltage. V, times

that component of the current that is in phase with the voltage,

cos

so that

cos

(single phase)

cos

(three phase)

where

generator output,

voltage,

current,

cos

power factor.

The reactive power,

PGR,

is computed by multipl>*ing the voltage, V, by the

other component of current, so that

PGR

sin

(single-phase situation) (9.5)

Frequently, reactive power,

PGR,

is defined as the quantity of power in ac circuits

obtained by taking the square root of the difference between the

square of the volt-

amperes, VI, and the square of

the

watts:

P,,

(VI cos

0)2

(9.6)

Reactive power is expressed as reactive volt-amperes or

vars.

The inductive reactance

is expressed as positive vars and capacitive reactance is expressed as negative vars.

The losses in the generator are given by the following equation:

cos

31,Ra

(9.7)

where

power loss for three-phase generator,

line-to-line voltage,

line current,

cos

power factor

armature current per phase if

connected

1, =I,

stator effective ac resistance per phase,

Note.

It is conventional that synchronous generators be rated by electri-

cos

V, volts

sin

cos

Figure

9.6

current

vector

diagram

showing lag and

powcr factor anglc.

amperes

Elementary Electrical Considerations Chap.

Synchronous Generators

157

120

100

,- ,-

friction losses

Load in

KVA

80% P.F.

Figure

9.7

Loss and efficiency for an

generator.

SOURCE:

U.S.

Bureau of

Reclamation.

cal output,

VI,

and expressed as kilovolt-amperes

(kVA),

not as kilowatts, un-

less a power factor is listed. In general, it is good to list it both ways to avoid

misunderstanding.

Power losses in ac generators are characterized by the following:

Copper losses

the armature and rotor windings

Frictior~ and windage losses

Core losses in the magnetic circuits of the generator

The copper losses are those losses due to resistance in

the windings of the armature

and rotor. The windage and friction losses are due to bearing friction and resistance

of the air on rotation of the rotor and are reasonably constant. The core losses are

due

to hysteresis and eddy currents in the cores and metal parts of both the rotor

and rhe stator. Figure

9.7

shows how the losses vary and gives a typical way in

which generator efficiency varies for a

5000-kVA generator. Further detail on the

technical aspects of generators can be found in books by Richardson

(1980)

and

Kosov

(1964).

EXCITATION

Ilnportant in electrical considerations for synchronous generators is the control of

power output. This can be accomplished in two ways. One

inethod of controlling

output is provided by changing the position of the wicket gates,

thus changing the

wster energy input to the turbine. The other method of power-output control is

the direct-current (dc) field excitation provided by a separate dc supply or

excitor.

Magnitude of generator power output

function of water flow

Magnitude of reactive power and sign are a function of

field excitation

(I,).

Bus

Magnitude of generator power output

a function of water flow

Induction generator must receive reactive power from

the system. Magnitude of

PR is a function

system

voltage.

Bus

Figure 9.8 Reactive and real power flow for synchronous and induction

generators.

Through this excitation there is an increase in field current and magnetic strength

by the rotor field poles. The increased magnetic field strength results in an in-

creased generator internal voltage. This will cause more reactive power to flow from

the generator armature to the electric system. Figure

9.8

shows a simple singe-

line diagram of how real power and reactive power flow from

synchronous

generator and an induction generator.

In the operation of a generator at different conditions of lagging current or

leading current,

the real power output and reactive power flow can be understood

by considering the reactive capacity curve of Fig.

9.9.

In this case it is seen that at

Real power (MW)

Figure 9.9 Representative reactive-capability curves for a hydropower generator.

158

Elementary ~lectricil Considerations Chap.

unity power factor there would be zero reactive power and the rated real power

output would be greatest. As the power factor decreases, the value of magnitude of

reactive power increases and the field current has to increase. As the power factor

reaches zero, the reactive power becomes a

maximum. In the case of leading current

situation for a synchronous generator the

same relative increase and reactive power

and decrease in real power output can be represented by a similar curve indicated in

lower right quadrant of a coordinate system. As the magnetic field current in-

creases, the reactive power increases.

Synchronous ac generators require that the field cores of the alternator

receive excitation from an external direct current (dc) source. The exciter is de-

signed to provide a flow of direct current to the rotor to produce the desired

voltage and var loading. The excitation systems in general use are:

Direct.co11nected shaft-driven dc generator, conventional or brushless

Separate prime mover or motor-driven dc generator

Alternating-current supply through static or mercury-arc rectifiers

Pilot excitation

Ma~n excitor

Commutator

Stator

Exciting coil

Pole

3-phase stator

winding

Rotor

ddd

ABC

Alternator terminals

Fiyre

9.10

Sinlplified diag~am of excitation ior a synchronous generator.

Induction Generators

159

Figure

9.10

shows a simplified diagram of how excitation is provided for

syn-

chronous generator. Sometimes the excitation system is designed to serve an

entire power plant with several turbines. At other times each turbine and generator

has

its own excitation equipment. Auxiliary power station needs for electrici'ty

should not be operated off the excitation equipment.

INDUCTION GENERATORS

An induction generator has no source of excitation and must draw reactive power

from the electrical system or transmission grid. The result of this fact is the ten-

dency to draw down the system voltage at the location of the generator. This is

then a disadvantage of induction generators. One means of overcoming this is to

provide a capacitor at the induction generator bus. The capacitors draw leading

FVND

DVNR

w.LvI

CLYIC-

.L.M

FROM

8US

lnduction generators may be used for

Operation in conjunction with a large system

Low capacity and high generator speeds

Applications not requiring power factor correction

Figure

9.11

Line diagram of electrical needs

for

induction generator. SOURCE:

Allis-Chalmers Corporation.

160

Elementary Electrical Considerations Chap.

current from the system, or in other words the capacitors are a source of lagging

current required by the induction generator.

unique requirement of the induc-

tion generator is that it requires lagging reactive power at all

times even when it is

not generating real power. Figure 9.8 shows in a comparative way in

simple

single-line diagram how real power flows away from the generator but reactive

power flows to the induction generator.

Induction generators are composed of a rotor with single squirrel-cage

un-

shielded windings. The stator can be standard induction motor design. Connection

to a bus or a system requires no excitation or synchronizing equipment. The

machine can be brought up to synchronous speed and the breaker closed to put

on line. Power factors for induction generators are higher at higher speeds, so the

tendency is to favor small units with higher speeds.

rule of thumb would be to

use induction generators on applications below

600

hp and above

600

rpm. Induc-

tion generators would not be useful in isolated small systems, in plants requiring

plant factor correction, in large-capacity plants, or in plants at the end of a system.

line diagram for an induction generator type of installation is shown in Fig. 9.1 1.

SPECIFYING GENERATOR EQUIPMENT

It is customary for the turbine manufacturer to work closely with the generator

manufacturer to furnish and specify the necessary engineering information for

generators. The various specified items of concern are listed in a checklist in Table

9.1.

TABLE 9.1 Checklist of Specification Items for an AC Generator

General considerations: number of units, inspection requirements, and working environ-

ment for erection, weight of major components, critical and outside dimensions

2. Rating characteristics: kilovolt-amperes, frequency phase, voltage, speed, amperes, and

power factor

Type: synchronous or induction, horizontal, inclined or vertical shaft, direct connected,

geared or belt drive, simple

gr complex load

Excitation: excitation voltage, requirements for direct-connected exciter, voltage con:

trol, and switch requirement

Temperature requirements: allowable temperature rise in degrees Celsius of stator, rotor,

rings, and bearing at critical load conditions, and temperature detectors

Connections and terminals: wye or delta connections, size of armature terminals, loca-

tion of armature terminals, needs for field terminals, and temperature-detection

terminals

Ventilation

and air-conditioning requirement: type of system, inlet and outlet locations

hfechanical requirements: number and types of bearings, weight carried by bearings, oil

and

water requirements, fire-protection needs, overspeed requirements, time to stop,

rotation sliaff size, and flywheel effect

Switching, Safety, and Electrical Control Equipment

Lower Paint Plant

7200

delta

phase. 150

KVA

480

delta

120

V.,

7.5

KVP.

tap

for station power

Line

)

switch

Start solenoid

stop solenoid

Incomplete sequence relay

Starting timer

Run

start

.#'

Reduced voltage

Reactor-type motor stater

100

kW,

514 RPM, Induction Generator

(Standard 150

H.P.

Squirrel

Cage Induction Motor)

Figure 9.12 Line diagram of electrical switching and control system for a very

small hydropower plant.

SOURCE:

Allis-Chalmers Corporation.

SWITCHING, SAFETY, AND ELECTRICAL CONTROL EQUIPMENT

The information on switch gear, equipment protection, and plant safety is very

specialized and is mentioned only briefly here. The equipment can vary from

simple to very complex, depending on the size of the hydroplant and whether the

plant is to be automatically or manually operated. Two line

diagrams showing the

electrical and control needs are presented in Figs.

9.12

and 9.13. The example

in Fig.

9.12

a small hydropower plant, Lower Paint Plant, of the Wisconsin-

hlichigan Power Company. Figure 9.13 shows a more sophisticated line diagram for

more conlplex medium-size hydro installation.

good reference for detail on de-

sign and planning for electrical switching, plant

protection,

electrical monitoring,

and transmission

is Brown (1970).

162

Elementary Electrical Considerations Chap.

Bus

~-+l$k

----+-

Generator

and synchronizing

potential

transformers

DEVICE SCHEDULE

51 V

Voltage controlled or restrained time

overcurrent relay

Time overcurrent relay

Instantaneous overcurrent relay

Power directional relay

40 Stator impedance or loss of field current

relay

Negative phase sequence current relay

64F Field circuit ground detector

Potential transformer failure relay

~ockout relay

IRI

Fixed or variable percent differential relay

Current-polarizd directional relay

-1-

----

Optional

Basedevice

Optional dwice

Figure 9.13

Line

diagram of electrical

switching

and control system

for

medium-sized hydropower plant.

Chap.

Prob!ems

REFERENCES

Brown,

G.;

Hydro-electric Engineering Practice,

Vol. 2, 2d ed. London: Blackie

Son, Ltd., 1970.

Kosov,

L.,

Electricui Machinery

and

Control.

Englewood Cliffs,

N.J.:

Prentice-

Hall, Inc., 1964.

Richardson,

V.,

Handbook of Rotating Electric Machinery.

Reston, Va.: Reston

Publishing Company, Inc., 1980.

PROBLEMS

9.1. Visit a hydroplant and obtain detail on as many electrical items as possible on

the checklist in Fig. 9.9.

9.2.

A three-phase 5000-kVA 13.8-kV 60-Hz 133.3-rpm Y-connected synchronous

generator for a hydro plant has losses as shown in Fig. 9.7. The per phase

winding reactance is 43.8

52.

If the generator is operating at 0.8 power factor

lagging, find:

(a) Electrical power output in

(b)

Reactive power in kvar

(c)

Input mechanical power

(d)

Line current in amperes

(e)

Effective resistance (assume that the rotor copper loss is 1% machine

rating, 50 kW)

(f)

Number of poles

(g)

Generator internal electromotive force

in volts on a line-to-line basis

(h) Percent voltage regulation (loaded to unloaded with constant field current)

9.3.

The terminal voltage of the machine of Problem 9.2 1s held constant by adjust-

ing field current as power factor goes to unity with the same output power.

Assume the same values of armature reactance (43.8

and resistance (0.2

part (e) of Problem 9.21. Find the generator internal voltage

on a line-to-

line basis. Find the percent voltage regulation.

RND

SPEED

REGULATION,

Sudden shutdowns of hydroelectric plants or changes in water flow through hy-

draulic turbines may cause problems ranging from rupture of penstocks due to

water

hammer to runner speed changes that cause the line current of the generators

to vary from the desired frequency. Regulating the water flow and coping with

slrJ,ic~i

closurc

ill'

gates

31id

s'alsqes require special equipment such as governors,

przsiure relief valves,

an11

suge tanks. Solving the problems of pressure control and

speed

regulation requires an understandi~lg of the basic theory of water hammer.

WATER HAMMER THEORY

AND

ANALYSIS

Water hammer is a phenomenon of pressure change

closed pipes caused when

flowing water in a pipeline is decelerated or accelerated by closing or opening a

valve or changing the velocity of the water rapidly in some other manner. The phe-

nomenon is accompanied by a series of positive and negative pressure waves

which

travel back and forth in the pipe system until they are damped out by friction. The

various

con~ponents of a hydropower installation must be capable of withstanding

these changes.

When a valve in a pipe or penstock carrying water is closed, the pressure head

imnlediately upstream of the valve is increased,

and

a pulse of high pressure is

propagated upstream to the nearest open water surface. On the downstream side of

the valve a lowered pressure moves in a downstream direction to the nearest open

water surface. If

the valve closure is rapid enough, a decrease in pressure may be

sufficient to cause a vapor pocket to form on the downstream side of the valve.

Water Hammer Theory and Analysis

165

When that vapor pocket collapses a high-pressure wave moves downstream.

The sequence 'of events following a sudden valve closure is presented

graplli-

cally in Fig. IO.l(a). When a valve in a pipeline is suddenly closed, the element of

water,

dE,

nearest the valve is compressed by the water flowing toward it and the

pipe is stretched

by the action. In the next time frame the element

dE7

is stopped

and it too is compressed. The water upstream of

dE2

continues to move at the

original velocity and successive elements of water are compressed.

The

action of

compression moves upstream as a wave until it reaches the open water surface and

the last element

dE,

is compressed and the entire conduit of water is under the

increased pressure head,

Ah.

The pressure pulse or wave moves at a velocity,

which is essentially the velocity of sound in water. Thus the pressure wave reaches

the open water surface in time,

Lla,

where

is the pipeline length from the

valve to the open water surface. At that time the kinetic energy of the moving water

has been converted to elastic energy in compressing the water and stretching the

pipe. At the open water surface the last element,

dE,

,,expands to its original state,

followed by other elements, causing a reverse or negative pressure wave. As this

wave travels downstream, conditions change

from zero velocity to a negative veloc-

ity of minus

and from the increased water pressure head,

Ah,

back to the

normal pressure head,

When the pressure wave reaches the valve, the pressure In

the pipeline has returned to normal and a time

2Lla

has elapsed. The water

moving away from the valve and back upstream now causes a reduction in pressure

and a negative pressure wave moves upstream to the open water surface. This

peri-

H.W.

Valve

Figure

10.1

Diagram of pressure wave action due to water hammer from sudden

valve closure.

Pressure Control and

Speed

Regulation Chap.

odic fluctuation is shown schematically in Fig. lO.l(b) as if the water did not have

friction acting. In reality, friction does act

within the water and

the boundaries

so that the pulses of pressure change have a decreasing

aniplitude as shown

Fig. lO.l(e).

Pressure Wave Velocity

Engineering analysis of water llamnler necessitates use of the term,

the

velocity of the pressure wave. According to

Parmakian (1955), the velocity of the

pressure wave in a pipe is given

the following formula:

where

velocity of the pressure wave, ft/sec

specific weight of water, 1b/ft3

acceleration of gravity, ft/sec2

=volume modulus of water

43.2

lo6 lb/ft2

diameter of pipe, in.

thickness of pipe, in.

Young's modulus of elasticity, 1b/ft2

for steel

4.32

lo9 1b/ft2

for cast iron

2.30

lo9 1b/ft2

for transite

0.49

lo9 1b/ft2

factor for anchorage and support of pipe

0.95 for pipe anchored at upper end and without expansion joints

0.91 for pipe anchored against longitudinal movzment

0.85 for pipe with expansion joints.

Parmakian

(1955) gives details on how to determine a value for the pressure wave

velocity,

for composite pipes and reinforced concrete pipes. If

penstock or pipe

embedded

Inass concrete, it is customary to consider the velocity of the pres-

sure wave,

equal to a value of less than

4660

ft/sec, the velocity of sound in

water. When

steel penstock is embedded

concrete, it is embedded with a corn-

pressible membrane, called a

nlastic

blanket,

around it or it is embedded under full

hydrostatic pressure plus the expected illcrease in pressure caused by water hammer.

Thus the steel penstock will expand under transient pressure to prevent internally

loading the

silrrounding concrete under tension stresses.

Two approaches to the theory of water

hammer have evolved: the rigid water

column theory

and the elastic water colunln theory.

Rigid Water Column Theory

Early investigators showed that for pipes that do not stretch, water that is

incompressible, the pipeline full

at all times, hydraulic losses negligible, velocity

uniform in the direction of tile pipe axis, a uniform pressure over the transverse

Water Hammer Theory and Analysis

167

cross section of the pipe, and no fluctuation in the open headwater level, the fol-

lowing equation applies for uniform rates of closure of the valve or gate:

where

(h,),,

maximum rise in pressure head at gate due to uniform closure, ft

initial steady pressure head at gate, ft.

LV'

=(rn)

where

length of pipe, ft

difference between initial and final steady velocity, ftlsec

acceleration of gravity, ft/sec2

time for gate closure or opening, sec.

The maximum drop in head at a gate or valve due to uniforrn gate opening is given

the equation

where

(h:),,

maximum drop in head at gate due to uniform gate opening, ft.

This approach is limited to slow-gate movement and according to Parmakian

(1955) should be used only when

is greater than L11000. For rapid gate move-

ments, the elastic water column theory and analysis should be used.

Elastic Water

Column

Theory

The physical significance of water hammer is shown diagrammatically in Fig.

10.2.

Parmakian

955) has shown that the following equations apply for the case

where the pipe is considered to deform under pressure increase due to water ham-

mer and the water is considered to be compressible. The equation of equilibrium

for the element of water is

and the equation of continuity is

where h

pressure head in pipe, ft or

velocity of water in pipe, ftlsec or mlsec

is considered positive mov-

ing from the

headwatcr)

acceleration of gravity, ft/sec2 or m/sec2