Chapter 6

Fundamentals ofHydraulic Transients

BAYARD E. BOSSERMAN IIWILLIAM A. HUNT

CONTRIBUTORSRobert C. GloverJoseph R. KroonM. Steve MerrillGary Z. Watters

The purpose of this chapter is to provide an overview

of the problems caused by hydraulic transients and an

insight into the circumstances that make a more thor-

ough analysis necessary. The fundamental theory of

hydraulic transient analysis is described simply, with

no attempt to present rigorous mathematical or ana-

lytical methods. Simple numerical examples include

surge pressure calculations, attenuation of surge pres-

sure by programmed valve closure, and the design of

pipe to resist upsurge and downsurge pressures.

For more complete discussions of hydraulic tran-

sients, see Parmakian [1], Rich [2], Wylie and Streeter

[3], Watters [4], and Chaudhry [5].

6-1. Introduction

Hydraulic transients are the time-varying phenomena

that follow when the equilibrium of steady flow in a

system is disturbed by a change of flow that occurs

over a relatively short time period. Transients are

important in hydraulic systems because they can

cause (1) rupture of pipe and pump casings; (2) pipe

collapse; (3) vibration; (4) excessive pipe displace-

ments, pipe-fitting, and support deformation and/or

failure; and (5) vapor cavity formation (cavitation,

column separation).

Someoftheprimarycauses (andfrequencyofoccur-

rence) of transients are (1) valve movements—closure

or opening (often), (2) flow demand changes (rarely),

(3) controlled pump shutdown (rarely), (4) pump fail-

ure (often), (5) pump start-up (rarely), (6) air venting

from lines (often), (7) failure of flow or pressure regu-

lators (rarely), and (8) pipe rupture (rarely).

The identification and calculation of pressures,

velocities, and other abnormal behavior resulting

from hydraulic transients make possible the effective

use of various control strategies, such as the

. selection of pipes and fittings to withstand the

anticipated pressures;

. selection and location of the proper control devices

to alleviate the adverse effects of transients; and

. identification of proper start-up, operation, and

shutdown procedures for the system.

The analysis of unsteady flow in pipe systems is

generally divided into two major categories.

Rigid water column theory (surge theory). The

fluid and pipe are inelastic, and pressure changes

propagate instantaneously. These flow conditions

are described by ordinary differential equations.

. Solution: closed-form integration or finite differ-

ence numerical integration.

6.1

. Advantages: the analysis can be applied by a person

with little numerical analysis skill and with limited

computational facilities.

. Disadvantages: the solutions, which are always ap-

proximations, are applicable only to simple pipe-

lines. Considerable experience is required to know

whether the results are applicable.

Elastic theory (water hammer). The elasticity of

both fluid and pipe affect the pressure changes. Pres-

sure changes propagate with wave speed, a, which

varies from about 300 to 1400 m/s (1000 to 4700 ft/s).

Flow conditions are described by nonlinear partial

differential equations.

. Solution: arithmetic, graphical method or method

of characteristics using finite difference techniques.

. Advantages: the theory accurately represents system

behavior and is, therefore, applicable to awide range

of problems. Pipe friction,minor losses, and varying

valve closure procedures can be incorporated.

. Disadvantages: applying the theory requires a sub-

stantial initial effort on the part of the user to learn

it, a digital computer programmed for the method

of characteristics, and a knowledge of the oper-

ational characteristics of system components to

set up the solution for the computer.

6-2. Nomenclature

In Chapters 6 and 7, ‘‘velocity’’ always means ve-

locity of water and ‘‘speed’’ means the velocity of

pressure waves. The symbols used in Chapters 6 and

7 are defined as follows.

a Elastic wave speed in water contained in a pipe

[in meters per second (feet per second)]

C Coefficient whose value depends on pipe

restraint

D Inside diameter of a pipe [in meters (inches or

feet)]

e Wall thickness of a pipe [in meters (inches or

feet)]

Ej Longitudinal joint efficiency in welded pipes

(dimensionless)

E Modulus of elasticity of pipe material [in new-

tons per square meter (pounds per square inch

or pounds per square foot); see Table 6-1].

g Acceleration due to gravity [in meters per

second squared (feet per second squared)]

h Head due only to surge [in meters (feet)]

Dh Change of head due to surge [in meters (feet)]

ha Allowable head due to surge [in meters (feet)]

K Bulk modulus of elasticity of the liquid [in new-

tons per square meter (pounds per square inch

or pounds per square foot); see Table A-8 or

A-9]

L Length of pipeline [in meters (feet)]

DP Change of pressure due to surge [in newtons per

square meter (pounds per square inch)]

DPa Allowable pressure change due to surge [in new-

tons per square meter (pounds per square inch)]

Patm Atmospheric pressure [in newtons per square

meter (pounds per square inch); see Table A-6

or A-7]

DPc Difference between external and internal pres-

sure on a pipe [in newtons per square meter

(pounds per square inch)]

Pv Vapor pressure of water [in newtons per square

meters (pounds per square inch); see Table A-8

or A-9]

SF Safety factor (dimensionless)

sy Yield stress [in newtons per square meter

(pounds per square inch)]

t Time (in seconds)

tc Critical time (2L/a; in seconds)

V Volume [in cubic meters (cubic feet)]

v Velocity [in meters per second (feet per second)];

in this chapter, ‘‘velocity’’ is average velocity of

fluid flow

Dv Change in velocity [in meters per second (feet

per second)]

Dva Allowable change in velocity [in meters per sec-

ond (feet per second)]

r Density [in kilograms per cubic meter (slugs per

foot; see Table A-8 or A-9)]

m Poisson’s ratio (dimensionless; see Table 6-1)

g Specific weight of water [in newtons per cubic

meter (pounds per cubic feet; see Table A-8 or

A-9)]

6-3. Methods of Analysis

Methods of analyzing pipelines for the effects of hy-

draulic transients, with or without various means of

controlling them or reducing the severity, can be

summarized as follows:

. Graphical [1]

. Arithmetic [2]

. Algebraic [3]

. Method of characteristics [3,4,5]

. Finite element

. Implicit differentiation

These are methods of analysis, not methods of

design. In analysis, the system is described mathemat-

ically, and the behavior of the system is predicted by

the analysis. In design, the desired physical results are

6.2 Chapter 6 Fundamentals of Hydraulic Transients

described and alternative means for attaining these

results are compared, which leads to a selection of

one or more control measures. Analysis should be

performed as a part of the design process.

All of the above methods involve the equations of

motion and continuity used to describe the velocity

and pressure variation in the pipeline. For computer

modeling, the most widely used is the method of

characteristics, in which the partial differential equa-

tions of motion and continuity are converted into

four first-order equations represented in finite differ-

ence form and solved simultaneously with a com-

puter. The method provides for

. the inclusion of many possible pipeline or system

features, such as junctions, pumping stations, air

chambers, air release valves, reservoirs, and line

valves;

. the inclusion of fluid friction;

. the retention of small or secondary terms in the

original equations so that accuracy is retained; and

. the computation of pressure and velocity as a func-

tion of time at various points throughout the entire

pipeline system.

6-4. Surge Concepts in Frictionless Flow

Water hammer can occur in a pipeline flowing full

when the flow is increased or decreased, such as when

the setting on a valve in the line is changed. When a

valve in a pipeline is closed rapidly, the pressure on

the upstream side of the valve increases, and the pulse

of increased pressure travels upstream at the elastic

wave speed, a. This pulse (called an ‘‘upsurge’’) de-

creases the velocity of flow. Downstream from an in-

line valve, the pressure is reduced and the wave of

decreased pressure travels downstream, also at the

elastic wave speed, a. This pulse (called a ‘‘down-

surge’’) decreases the velocity of flow. If the velocity

is reduced too rapidly and the steady-state pressure is

low enough, the downstream pressure can be reduced

to vapor pressure, which creates a vapor pocket. A

large vapor pocket (called ‘‘column separation’’) can

collapse with a dangerous explosive force produced

by the impact between solid water columns and can

cause the pipe to burst. This phenomenon can also

occur upstream of the valve when the reflected posi-

tive wave returns to the valve.

The events following a sudden closure of a valve

located a distance (L) downstream from a reservoir is

described in Figure 6-1. Friction is neglected, and the

energy gradeline (EGL) and hydraulic gradeline

(HGL) are assumed to coincide because velocity

heads are small compared with water hammer pres-

sure heads. The steady-state EGL is calledH, and the

added-pressure head pulse is called h. The velocity of

the fluid under steady-state conditions is vo just be-

fore the valve is closed (at t ¼ 0).

The sequence of events between the valve and the

reservoir occurs in a four-phase cycle; the duration of

each phase is the time for the pressure wave to travel

between the valve and the reservoir (the length of the

pipeline divided by the elastic wave speed, L/a). The

sequence occurs as follows:

Table 6-1. Physical Properties of Pipe Materials

Modulus of elasticity

U.S. customary units

Material Poisson’s ratio SI units (N=m2) lb=in:2 lb=ft2

Aluminum 0.33 7.30 E þ 10 1.05 E þ 7 1.51 E þ 9

Asbestos-cement 0.30 2.30 E þ 10 3.40 E þ 6 4.90 E þ 8

Brass 0.34 1.03 E þ 11 1.50 E þ 7 2.16 E þ 9

Copper 0.30 1.10 E þ 11 1.60 E þ 7 2.30 E þ 9

Ductile iron 0.28 1.66 E þ 11 2.40 E þ 7 3.46 E þ 9

Gray cast iron 0.28 1.03 E þ 11 1.50 E þ 7 2.16 E þ 9

HDPE 0.45 1.0 E þ 9a 1.5 E þ 5a 2.2 E þ 7a

PVC 0.45 2.70 E þ 9 4.00 E þ 5 5.76 E þ 7

Steel 0.30 2.07 E þ 11 3.00 E þ 7 4.32 E þ 0

Concrete — 4:73� 106ffiffiffiffiffiffif 0c

pb 57,000

ffiffiffiffiffiffif 0c

pc

aAt 168C (608F). Increases greatly with decreasing temperature and vice versa.

bf 0c is ultimate strength in newtons per square meter.

cf 0c is ultimate strength in pounds per square inch.

6-4. Surge Concepts in Frictionless Flow 6.3

1. 0 # t # L=a. At t ¼ 0, the fluid just upstream

from the valve is compressed and brought to rest.

Part of the pipe (section BC) is expanded and

stretched, as shown in Figure 6-1a. This process is

repeated for each successive increment of fluid as the

pressure wave travels upstream. The fluid upstream

from the wave front continues to flow downstream

until it is stopped by the advancing pressure wave

front. When the pressure wave reaches the reservoir

at t ¼ L=a, the fluid (at rest in the pipe) is under a

total pressure head H þ h, or h greater than the static

head in the reservoir.

2. L=a # t # 2L=a. The pressure head difference

at t ¼ L=a at the reservoir causes the fluid to flow

from the pipe back into the reservoir with a velocity

�vo. The pressure along AB is reduced to the original

steady-state level, H, and a negative wave producing

normal pressure propagates back to the valve, as

shown in Figure 6-1b. At t ¼ 2L=a, the pressure is

normal (equal to H) along the pipe but the velocity

throughout the pipe is negative; that is, water is flow-

ing away from the valve.

3. 2L=a # t # 3L=a. At t ¼ 2L=a, there is no fluidavailable to maintain the upstream flow at the valve,

and the normal pressure head, H, is reduced by h to

bring the fluid in section BC to rest (Figure 6-1c). This

wave of reduced pressure propagates back toward the

reservoir, all fluid comes to rest, and the reduced

pressure allows the fluid to expand and the pipe walls

to contract. At t ¼ 3L=a, the reduced pressure, H – h,

exists all along the pipe, the velocity is zero throughout

the pipe, and the static pressure head in the pipe is less

than the pressure head in the reservoir.

4. 3L=a # t # 4L=a. At t ¼ 3L=a, the unbal-

anced pressure head at the reservoir causes the fluid

to flow back into the pipe at a velocityþvo. A wave of

pressure at the original static pressure head level

propagates downstream toward the valve, but in sec-

tion BC, the pressure is still reduced to H – h (Figure

6-1d). At t ¼ 4L=a, the pressure throughout the pipeis normal (equal to H), and the velocity is the same as

the original þvo prior to the valve closure. But when

the velocity, vo, reaches the valve, the four-phase

cycle repeats and continues to repeat periodically

every 4L=a time period.

The variation of pressure with time during a 4L/a

interval is shown at several points along the pipeline in

Figure 6-2. Friction in a real pipe system eventually

dampens the increasedwater hammer pressure head, h.

The principal concepts indicated by the events

following a sudden valve closure are as follows:

. The time L/a is a significant parameter for water

hammer analysis.

Figure 6-1. Sequence of events for one cycle after sudden valve closure.

6.4 Chapter 6 Fundamentals of Hydraulic Transients

. The time 2L/a is critical because the pressure head

at the valve reaches a maximum at 2L/a. A valve

closed in any shorter time produces the same max-

imum pressure head rise at the valve, whereas the

pressure rise is reduced if the valve is closed in a

longer time interval. Hence,

tc ¼ 2L

a(6-1)

where tc is the critical time, L is the length of the pipe,

and a is the elastic wave speed.

Pressure Head Change

A momentum analysis of the flow conditions for the

valve closure shows the pressure head change, Dh, is afunction of the change in flow, Dv.

Dh ¼ �aDv

g(6-2)

where Dh is the change in pressure head in meters

(feet), a is elastic wave speed in meters per second

(feet per second), Dv is the change in velocity (of

water) caused by the event in meters per second

(feet per second), and g is the acceleration due to

gravity in meters per second squared (feet per second

squared). Use the negative sign for waves traveling

upstream, use the positive sign for waves traveling

downstream, and note that Dv ¼ v2 � v1, where v1 is

the velocity prior to the change in flow rate and v2is the velocity following the change. If the flow is

suddenly stopped, Dv ¼ v1 and Dh ¼ av1=g. Note,

too, that Dh is positive if Dv is negative. If the valve

on the downstream end of a pipe is closed incremen-

tally, Equation 6-2 becomes

Figure 6-2. Head fluctuations at selected points in the pipeline of Figure 6-1 after valve closure.

6-4. Surge Concepts in Frictionless Flow 6.5

SDh ¼ � a

gSDv (for t < tc) (6-3)

Transient pressure heads due to valve closure can

be reduced by slowly closing the valve over a time

interval greater than tc, as discussed in Section 6-5.

Elastic Wave Speed

The pressure head change due to a change in the flow

rate requires the calculation of the elastic wave speed

in the pipe. The wave speed depends on both the fluid

properties and the pipe characteristics.

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K=r

1þ CK

E

� �D

e

� �vuuut (6-4)

where K is the bulk modulus of elasticity of the liquid

in newtons per square meter (pounds per square

foot), E is the modulus of elasticity of pipe material

in newtons per square meter (pounds per square

foot), D is inside pipe diameter in meters (feet), e is

the pipe wall thickness in meters (feet), C is a correc-

tion factor for type of pipe restraint, and r is the

density of the fluid in kilograms per cubic meter

(slugs per cubic foot). The bulk modulus given in

Table A-8 must be changed from kilopascals to new-

tons per square meter, and in Table A-9 it must be

converted from pounds per square inch to pounds per

square foot.

For thin-walled pipes (those with D=e > 40), the

correction factor, C, varies according to pipe re-

straints. Values for three cases are

. C1 ¼ 1:25� m for pipes anchored at the upstream

end only (Wylie and Streeter [3] use 1:0� m=2; thedifference has little practical significance);

. C2 ¼ 1:0� m2 for pipes anchored against axial

movement; and

. C3 ¼ 1:0 for pipes with expansion joints through-

out.

The symbol m is Poisson’s ratio (see Table 6-1 for

properties of pipe materials). Expressions for C3 for

thick-walled pipes (D=e < 40), which are more com-

plex, are given by Wylie and Streeter [3]. Buried pipes

are best represented by the factor C2. Typical values

for wave speed for water in pipes are given in Table

6-2. Because the elastic wave speed, a, for steel and

ductile iron pipe is often close to 980 m/s (3200 ft/s),

the change in pressure head from Equation 6-2 is,

roughly, �100v. For PVC, h ¼ �34v.

The speed of the pressure wave in a pipeline carry-

ing liquids is greatly reduced if bubbles of free gas are

entrained, as shown in Table 6-3. A detailed discus-

sion of this phenomenon is given by Wylie and

Table 6-2. Typical Wave Velocities in Pipe for WaterContaining Dissolved Air

Wave velocities

Pipe material m/s ft/s

Asbestos cement 820–1200 2700–3900

Copper 1000–1300 3400–4400

Ductile iron 980–1400 3200–4500

HDPEa 180–370 600–1200

PVC 300–600 1000–2000

Steel 600–1200 2000–4000

a For a modulus of elasticity of 1:03Eþ 9N=m2(150,000 lb=in:2).

Table 6-3. Effect of Air Entrainment on Wave Speeda

Wave speed

Vair=Vtotal m/s ft/s

0 1200 4000

0.001 610 2000

0.002 460 1500

0.004 300 1000

0.008 210 700

aAfter Wylie and Streeter [3].

Example 6-1Effect of Pipe on Wave Speed and Pressure

Problem: For 300-mm (12-in.) pipes of ductile iron, steel, and PVC, find the effects of pipe

material, bedding, and joint conditions on wave speed and water hammer pressure. Assume the

pipes are 3000 m (9840 ft) long and carry 0:15m3=s (5:3 ft3=s) of water at a temperature of

158C (598F).Solution: The wave speeds are based on Equation 6-4. The calculation of wave speed to the

nearest 30 m/s (100 ft/s) is usually sufficient, because the accuracy of the data does not justify

greater precision. The critical valve closure times, tc, are based on Equation 6-1. Valve closure

6.6 Chapter 6 Fundamentals of Hydraulic Transients

Streeter [3]. There seems to be no way of predicting

air volume entrainment. Neglecting the effect of air

entrainment provides a conservative analysis. How-

ever, as the wave speed decreases, L/a increases and

the time for closure of valves must be increased.

6-5. Slow Closure of Valves

To limit the pressure rise, the maximum deceleration

of the water during the critical time period, tc, must

be limited. The maximum allowable deceleration can

be calculated by using the ratio

DPa

DP¼ Dha

Dh¼ Dva

Dv(6-5)

where DP is pressure rise, Dh is the head rise, Dv is

the change in velocity, and the subscript, a, means

allowable. The terms in the denominator are for in-

stantaneous valve closure.

Manufacturers can provide either (1) curves of the

valve headloss coefficient, K, in Equation 3-15

wherein h ¼ Kv2=2g (shown for a butterfly valve in

Figure 6-3) or (2) values of flow coefficients, Cv,

wherein Cv equals water flow in gallons/minute at

708F and 1 lb=in:2 pressure differential shown for

ball and eccentric plug valves in Figure 6-4 (see

Appendix A for conversion to SI units). Both K and

Cv vary greatly with valve opening, so either is a

convenient aid for solving valve problems.

Equation 6-5 can lead either to simplified equa-

tions or to computer solutions for determining the

allowable rate of valve closure so that a given pres-

sure is not exceeded. To be valid, a method of solu-

tion must include

. the change of valve coefficient, K, in the formula

for headloss (Equation 3-15), or

. the shape of the valve closure curve, Cv=Cvo, and

the nonlinear action of some valve stems, and

. the effect of the increasing pressure, which tends to

sustain the original flow through the valve. The

pressure, for example, depends on the shape of an

associated pump curve and the location of the

valve as well as on other characteristics of the

system.

Because resistance to flow is relatively small in the

initial stages of closure, a valve can be quickly closed

to near shut-off if the final closure is slow. Of the

three gate valve closure programs occurring in the

same interval, 3tc, as shown in Figure 6-5, the least

pressure is generated by a 95% closure in 0:5tc with

the remainder of the closure occurring in 2:5tc.A valve poorly selected with respect to Cv closure

characteristics can often be the difference between a

in any time interval less than tc subjects the pipe to the maximum pressure, Dh, given by

Equation 6-2. Note that K ¼ 2:15� 109 N=m2(4:49� 107 lb=ft2) for water.

SI Units U.S. Customary Units

Ductile iron Steel PVC Ductile iron Steel PVC

OD [m (ft)] 0.335 0.324 0.324 1.10 1.063 1.063

e [m (ft)] 0.0094 0.0048 0.0149 0.0308 0.0157 0.0259

E [N=m2 (16=ft2)] 1:66� 1011 2:70� 1011 2:70� 109 3:46� 109 4:32� 109 5:76� 107

m 0.28 0.30 0.45 0.28 0.30 0.45

C1 ¼ 1:25� m 0.97 0.95 0.80 0.97 0.95 0.80

C2 ¼ 1� m2 0.92 0.91 0.80 0.92 0.91 0.80

C3 ¼ 1:0 1.00 1.00 1.00 1.00 1.00 1.00

Wave speed, a [m/s (ft/s)], from Equation 6-4:

C1 ¼ 1:25� m, no air 1220 1150 280 4000 3770 930

C2 ¼ 1� m2, no air 1210 1130 280 3970 3710 930

C3 ¼ 1:0, no air 1190 1110 260 3920 3640 840

C3 ¼ 1:0, 0.2% air 490 490 230 1620 1600 760

For C3 ¼ 1:0 and no air:

tc ¼ 2L=a (in s) 5.0 5.4 23 5.0 5.4 23

Flow [m/s (ft/s)] 1.8 1.8 1.9 5.9 6.0 6.2

h [in m (ft)] 219 206 49.4 718 676 162

P [in kPa (lb=in:2)] 2150 2020 480 310 290 70

6-5. Slow Closure of Valves 6.7

problem situation and no problem at all. Ball valves

(with their worm and compound lever actuators)

have the ideal characteristic of closing rapidly at

first followed by very slow closure without any com-

plicated control gear. The capacity of the valve is

reduced 80% at about 20% closure (see Figure 5-2).

Figure 6-3. Headloss coefficient for a butterfly valve.

Figure 6-4. Valve closure functions. (a) Plug valve (DeZurik Series 100), the plug angle is linear with the time ofclosure; (b) ball valve (Williamette List 36), actuator travel is linear with the time of closure.

6.8 Chapter 6 Fundamentals of Hydraulic Transients

In contrast, the eccentric plug with linear valve stem

angle (or percentage of closure) has a near linear

characteristic and would require either a much longer

time for closure or a programmed closure to equal the

ball valve’s control of surge pressure.

As a rule, valves should not be closed in less than 2

to 10 times tc. Insist on a computer solution wherever

more certainty than this is needed, or when the prob-

lem is important because of the size of the pipe, the

amount of flow, or the dynamic pressure involved.

Example 6-2Seating a Valve

Problem: A 200-mm (actually, a 203-mm or 8-in.) butterfly valve discharges a flow of 12.6 L/s

(200 gal/min) to atmosphere from a level pipe 4.83 km (3 mi) long with a gauge pressure of

690 kPa (100 lb=in:2) just upstream of the valve. The valve design is such that it will close from

its fully open position in 50 s.

Find the initial valve position (percentage of opening) and estimate the surge pressure when

the valve closes.

Solution: First, find the percentage of valve opening from Equation 3-15 and a manufacturer’s

curve of K as shown in Figure 6-3. Then find the approach velocity in the pipe.

SI Units U.S. Customary Units

v ¼ Q

A¼ 0:0126m3=s

(p=4)(0:200)2¼ 0:401m=s

v ¼ Q

A¼ 2:00 gal=min

(p=4)(8=12)2

" #

0:00223 ft3=s

gal=min

� �¼ 1:28 ft=s

The differential head across the valve is (see Tables A-8 and A-9):

h ¼ P

g¼ 690,000N=m2

9789N=m3¼ 70:5m h ¼ P

g¼ 100 lb=in:2

62:3 lb=ft3� 144 in:2

ft2¼ 231 ft

Figure 6-5. Water hammer caused by various gate valve closure programs. After Watters [4, p. 195].

6-5. Slow Closure of Valves 6.9

The assumption of a steady pressure upstream

from the valve is often erroneous. The pressure usu-

ally changes with a change in flow as the system

follows the characteristic pump H-Q curve. For

pumps with type numbers greater than about 85 (spe-

cific speeds greater than about 4400), the head devel-

oped by a pump increases substantially as the flow

decreases. If the type number is about 125 (specific

speeds about 6600) or more, the increase in head is

dramatic, as shown in Figure 10-16.

If the time of closure in Example 6-2 were greater

than tc, the resulting pressure surge would be less

than the 40 m (130 ft) calculated, and a computer

analysis would be used to reveal the pressure rise.

Note that the shape of the valve closure curve

would be very important in such an analysis.

6-6. Surge Concepts in Flow with Friction

The explanation of water hammer in Section 6-4 is

simplified by omitting friction. Such simplification

can be justified in some systems, such as those with

short force mains in which friction head is a small

part of the total design head. In most systems, how-

ever, friction contributes substantially to the total

head, and it is important to understand its effect.

Figure 6-6 is similar to Figure 6-1a except for the

hydraulic grade line, which slopes during steady-state

flow. Shortly after sudden closure of the valve at

point O, the wave front would arrive at point A.

Consider the anomalies that would ensue by follow-

ing the construction shown for frictionless flow in

Figure 6-1a in which the pressure rise for the wave

front at B forms rectangle abcd. Its counterpart is

trapezoid oabc in Figure 6-6, but because the hy-

draulic grade line bc is sloping, water continues to

flow past point A (corresponding to point B in Figure

6-1); therefore, v (and, hence, h) would be less in

Figure 6-6 than in Figure 6-1a for the same original

velocity. But flow continues only until the HGL be-

comes level. Meanwhile the wave front progresses to

point B (Figure 6-6) and the same phenomenon oc-

curs again, with more water flowing past point A, the

SI Units U.S. Customary Units

From Equation 3-16

K ¼ 2gh

v2¼ 2� 9:81� 70:5

(0:401)2¼ 8600 K ¼ 2gh

v2¼ 2� 32:2� 231

(1:28)2¼ 9080

The discrepancy between SI and U.S. units is due only to rounding off the difference between

200 mm and 8 in. From Figure 6-3, the valve is open about 78 (nearly closed).

The wave speed can be calculated from Equation 6-4 or simply estimated as about 100 g.

For linear valve closure, the last 78 of valve movement would require (7=90)50 ¼ 3:9 s.

a ¼ 100� 9:81 � (980) a ¼ 100� 32:2 � (3200)

tc ¼ 2L

a¼ 2� 4830

980� 10 s tc ¼ 2L

a¼ 2� 3� 5280

3200� 10 s

Because the valve will close in 3.9 s (less than the critical time, 10 s), it closes quickly enough to

generate the full transient pressure given by Equation 6-2.

Dh ¼ aDv

g¼ 980� 0:401

9:81� 40m Dh ¼ aDv

g¼ 3200� 1:28

32:2� 130 ft

And the total pressure in the pipeline is

P ¼ 690 kPaþ 40� 9:81 ¼ 1080 kPaP ¼ 100 lb=in:2 þ 130� 0:433

lb=in:2

ft

¼ 156 lb=in:2

6.10 Chapter 6 Fundamentals of Hydraulic Transients

pressure rising, the pipe expanding, and the water

compressing. Wylie and Streeter [3] call this occur-

rence ‘‘packing.’’ As the wave front approaches the

reservoir, friction decreases v and h, a phenomenon

called ‘‘attenuation.’’ Eventually, the pressure builds

to a maximum above the level of the reservoir and

finally dies to the reservoir level.

Depending on the length of the pipeline, the pres-

sure surges at the valve might appear, as shown in

Figure 6-7. With each reversal of the pressure wave,

friction decreases the pressure changes, so pressure

eventually coincides with the reservoir level.

6-7. Column Separation

In the preceding sections, the pipelines are assumed to

be level, but in real systems pipes may slope and down-

surges result when power fails or when valves at the

upstream end (the usual configuration) close quickly.

Under some conditions, the downsurges can cause

column separation—a condition to be avoided at any

cost by a proper control strategy. As shown in Figure

6-8, a knee (a reduction in gradient or a change from a

positive to a negative gradient) makes a pipeline espe-

cially vulnerable. If the power fails, the pumps stop

quickly with an effect like closing a valve. The up-

stream flow (near the pumping station) stops whereas,

due to inertia, flow continues at the downstream end

(near the discharge). The static hydraulic gradeline

begins to decay as shown by successive curves labeled

t ¼ 1 s, t ¼ 2 s, t ¼ 3 s, until, at t ¼ 4 s, a slight nega-

tive pressure occurs between the pump and the knee.

At t ¼ 4:5 s, vapor pressure exists over a considerable

length of the pipeline, and the water is boiling and

forming large pockets of vapor. Column separation

has occurred. On the upsurge of pressure that follows,

the vapor pockets collapse and the two liquid columns

can come together at literally express-train speed.

Since the water is almost incompressible, the forces

at impact can be enormous.

The knee makes the situation in Figure 6-8 worse,

but note that column separation would occur with or

without the knee and would occur even if the pipeline

hadauniformgradient.However, if thepipelineprofile

were flatnear thepumpand the steepgradientoccurred

near the reservoir, columnseparationmightbeavoided

—a useful control strategy that is maintenance free.

Figure 6-6. Packing and attenuation in a long pipe with friction.

Figure 6-7. Surges at a valve in a pipe with friction.

6-7. Column Separation 6.11

Acomputer analysis of a similar problem is givenby

Watters [4, pp. 235 ff], and the results of still another

somewhat similar problem with and without surge

control devices are shown in Figures 7-12 and 7-13.

The hydraulic gradelines after power failure can

sometimes be crudely estimated at several time inter-

vals if thedecelerationtimeof thepumpisknownorcan

be estimated. Flow through centrifugal pumps after

power failure is a functionofmany variables, including

. inertia and speed of the pump, driver, and water

within the casing;

. length and profile of the pipe;

. steady-state hydraulic gradeline;

. velocity of flow; and

. suction conditions in the wet well.

The true shape of the hydraulic gradelines requires

solution by a computer.

6-8. Criteria for Conducting TransientAnalysis

Every pump and pipeline system is subject to transi-

ent pressures, but in practice, it is impractical to

spend the time and expense necessary to analyze all

of them. The following empirical guidelines, which

seem to be satisfactory in most (though certainly

not all) situations, can be used to decide whether a

complete transient analysis is required.

Do Not Analyze

. Pumping systems with flow less than 23m3=h(100 gal/min). Discharge piping is usually such

that velocity is low and transient pressures are

low. Even if transient pressures are high, small

diameter (100-mm or 4-in.) piping has a high pres-

sure rating and can usually withstand the pressures.

. Pipelines in which the velocity is less than 0.6 m/s

(2 ft/s).

. Distribution systems or pipe networks (as in

community potable water systems). The many

junctions significantly dissipate the pressure

waves.

. Reciprocating pumps, because virtually every re-

ciprocating pump should have a pulsation damp-

ener on the discharge (see Ekstrum [7] for methods

of sizing such dampeners).

. Pumping systems with a static differential pressure

between suction and discharge of less than about

9 m (30 ft).

Warning: it is possible that a very low static head

coupled with a relatively high dynamic head could

result in a column separation problem.

Figure 6-8. Successive hydraulic gradelines following power failure. Adapted from Watters [4, p. 271].

6.12 Chapter 6 Fundamentals of Hydraulic Transients

Do Analyze

. Pumping systems with a total dynamic head greater

than 14 m (50 ft) if the flow is greater than about

115m3=h (500 gal/min).

. High-lift pumping systems with a check valve, be-

cause high surge pressures may result if the check

valve slams shut upon flow reversal.

. Any system in which column separation can occur:

(1) systems with ‘‘knees’’ (high points), (2) a force

main that needs automatic air venting or air

vacuum valves, or (3) a pipeline with a long [more

than 100 m (300 ft)], steep gradient followed by a

long, shallow gradient.

. Someconsultants analyze any forcemain larger than

200 mm (8 in.) when longer than 300 m (1000 ft).

Checklist

Some additional insight can be gained from the fol-

lowing conditions, which tend to indicate the serious-

ness of surge in systems with motor-driven centrifugal

pumps. A serious surge may well occur if any one of

these conditions exists. If two or more conditions

exist, a surge will probably occur with a severity

proportional to the number of conditions met [8–11].

. There are high spots in pipe profile

. There is a steep gradient: length of force main is

less than 20 TDH

. Flow velocity is in excess of 1.2 m/s (4 ft/s)

. Factor of safety (based on ultimate strength) of

pipe (and valve and pump casings) is less than 3.5

for normal operating pressure

. There can be slowdown and reversal of flow in less

than tc. There is check valve closure in less than tc. There is any valve closure (or opening) in less than

10 s

. There can be damage to pump and motor if

allowed to run backward at full speed

. Pump can stop or speed can be reduced to the point

where the shut-off head is less than static head

before the discharge valve is fully closed

. Pump can be started with discharge valve open

. There are booster stations that depend on oper-

ation of main pumping station

. There are quick-closing automatic valves that be-

come inoperative if power fails or pumping system

pressure fails.

Criteria for determining whether to use simple

hand calculations or a more detailed computer pro-

gram are also given in Pipeline Design for Water and

Wastewater [8, p. 65].

Shut-downs will occur, so plan for them. They can

result in low pressures and column separation at

knees in steep pipelines. Air venting valves that

close too rapidly while an empty pipe is being filled

also cause destructive hydraulic transients. Even on

low-lift pumping stations, depending on the pipe pro-

file, column separation can occur in the vicinity of the

discharge header or farther downstream.

Computers

There is no simple, easy way to perform reliable

transient analyses. Computer modeling is the most

effective means available, but there are practical con-

straints on time and cost. Both computer time and

the labor needed to analyze and review are expensive,

so the extent of the analysis should be related to the

size and cost of the project. For example, spending

$1000 to $5000 to analyze transients for a $1,000,000

project is probably worthwhile even if no problem is

found.

6-9. References

1. Parmakian, I., Waterhammer Analysis, Dover, New

York (1963).

2. Rich, G. R., Hydraulic Transients, Dover, New York

(1963).

3. Wylie, E. B., and V. L. Streeter, Fluid Transients, Feb

Press, Ann Arbor, MI (1982, corrected copy 1983).

4. Watters, G. Z., Analysis and Control of Unsteady Flow in

Pipelines, 2nd ed., Butterworths, Stoneham, MA (1984).

5. Chaudhry, M. H., Applied Hydraulic Transients, Van

Nostrand Reinhold, New York (1979).

6. Wood, D. J., and S. E. Jones, ‘‘Water hammer charts for

various types of valves,’’ Journal of the Hydraulics Div-

ision, Proceedings of the American Society of Civil En-

gineers, 167–178 (January 1973).

7. Ekstrum, J. D., ‘‘Sizing pulsation dampeners for recipro-

cating pumps,’’Chemical Engineering (January 12, 1981).

8. Pipeline Design for Water and Wastewater, American

Society of Civil Engineers, New York (1975).

9. AWWA M11, Steel Pipe—A Guide for Design and In-

stallation, p. 54, American Water Works Association,

Denver, CO (1985).

10. Kerr, S. L., ‘‘Minimizing service interruptions due to

transmission line failures: Discussion,’’ Journal of the

American Water Works Association, 41, 634 (July 1949).

11. Kerr, S. L., ‘‘Water hammer control,’’ Journal of the

American Water Works Association, 43, 985–999 (De-

cember 1951).

6-9. References 6.13