PIPELINEDESlIiNFOR WATER ENMNEERS
DEVELOPMENTS I N WATER SCIENCE, 15 advisory editor VEN T E CHOW Professor of Civil and Hydrosystems Engineering, Hydrosystems Laboratory, University of Illinois, Urbana, Ill., U.S.A. OTHER TITLES I N THIS SERIES
1 G. BUGLIARELLO AND F. GUNTER COMPUTER SYSTEMS AND WATER RESOURCES
2 H.L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY
3 Y.Y. HAIMES, W.A. HALL AND H.T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADEOFF METHOD
4 J.J. FRIED GROUNDWATER POLLUTION
5 N. RAJARATNAM TURBULENT JETS
6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (see Vol. 15)
7 V. HI~LEK AND J.
SVEC
GROUNDWATER HYDRAULICS
8 J. BALEK HYDROLOGY AND WATER RESOURCES I N TROPICAL AFRICA
9 Th. A. McMAHON AND R.G. MElN RESERVOIR CAPACITY AND YIELD
10 G. KOVACS SEEPAGE HYDRAULICS
11 W.H. GRAF AND C.H. MORTIMER (EDITORS) HYDRODYNAMICS OF LAKES
12 W. BACK AND D.A. STEPHENSON (EDITORS) CONTEMPORARYHYDROGEOLOGY
1 3 M.A. M A R I ~ OAND J.N. LUTHIN SEEPAGE AND GROUNDWATER
14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE
15 D. STE,PHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series)
WATER ENGINEERS Professor of Hydmulic Engineering, University of the Witwatersrand Johannesburg, South Africa Consulting Engineer
SECOND EDITION (COMPLETELY REVISED EDITION OF VOL. 6 IN THE SERIES)
ELSEVIER SCIENTIFIC PUBLISHING COMPANY 1981 Amsterdam  Oxford  New York
ELSEVIER SCIENTIFIC PUBLISHING COMPANY 1, Molenwerf P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada:
ELSEVIERINORTHHOLLAND INC. 52, Vandehilt Avenue New York, N.Y. 10017
First edition 1976 Second impression (with amendments) 1979 Second edition (Completely revised) 1981
Library
of Congress Cataloging i n Publication Uata
Stephenson, David, 1943Pipsline design f o r water engineers. Includes bibliographies and indexes. 1. Waterpipes. I. Title. 628.1'5 TD49l.S743 1981 ISBN 0444419918
815508 MCF2
ISBN: 0444419918 (Val. 15) ISBN: 0444416692 (Series) 0 Elsevier Scientific Publishing Company, 1981 All rights reserved. No part of this publication may be reproduced, stored i n a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330,1000 A H Amsterdam, The Netherlands Printed in The Netherlands
V
Pipelines are being constructed in everincreasing diameters, lengths and working pressures. Accurate and rational design bases are essential to achieve economic and safe designs. Engineers have for years resorted to semiempirical design formulae. Much work has recently been done in an effort to rationalize the design of pipelines. This book collates published material on rational design methods as well as presenting some new techniques and data. Although retaining conventional approaches in many instances, the aim of the book is to bring the most modern design techniques to the civil or hydraulic engineer. It is suitable as an introduction to the subject but also contains data on the most advanced techniques in the field. Because of the sound theoretical background the book will also be useful to undergraduate and postgraduate students. Many of the subjects, such as mathematical optimization, are still in their infancy and the book may provide leads for further research. The methods of solution proposed for many problems bear in mind the modern acceptance of computers and calculators and many of the graphs in the book were prepared with the assistance of computers. The first half of this book is concerned with hydraulics and planning of pipelines. In the second half, structural design and ancillary'features are discussed. The book does not deal in detail with manufacture, laying and operation, nor should it replace design codes of practice from the engineer's desk. Emphasis is on the design of large pipelines as opposed to industrial and domestic piping which are covered in other publications. Although directed at the water engineer, this book will be of use to engineers involved in the piping of many other fluids as well as solids and gases. It should be noted that some of the designs and techniques described may be covered by patents. These include types of prestressed concrete pipes, methods of stiffening pipes and branches and various coatings. The S.I. system of metric units is preferred in the book although imperial units are given in brackets in many instances. Most graphs and equations are represented in universal dimensionless form. Worked examples are given for many problems and the reader is advised to work through these as they often elaborate on ideas not highlighted in the text. The algebraic symbols used in each chapter are sumarized at the end of that chapter together with specific and general references arranged in the order of the subject matter in the chapter. The appendix gives further references and standards and other useful data.
vi
PREFACE TO SECOND EDITION
The gratifying response to the first edition of this book resulted in small amendments to the second impression, and some major alterations in this new edit ion. The chapters on transport of solids and sewers have been replaced by data more relevant to water engineers. Thus a new chapter on the effects of air in water pipes is included, as well as a chapter on pumping systems f o r water pipelines. The latter was reviewed by Bill Glass who added many of his own ideas. There are additions and updating throughout. There is additional information on pipeline economics and optimum diameters in Chapter 1. A comparison of currently used friction formulae is now made in Chapter 2 . The sections on noncircular pipe and partly full pipes and sewer flow are omitted.
These are largely of in
terest to the drainage engineer and as such are covered in the author's book 'Stormwater Hydrology and Drainage' (Elsevier,l981). A basic introduction to water hammer theory preceeds the design of water hammer protection of pumping and gravity lines in Chapter 4 . The sections on structural design of flexible pipes are brought together. An enlarged section on soilpipe interaction and limit states of flexible pipes preceeds the design of stiffened pipes. Although some of the new edition is now fairly basic, it is recognised that this is desirable for both the practicing engineer who needs refreshing and the student who comes across the problem of pipeline design for the first time.
v i i
ACKNOWLEDGEMENTS
The basis for this book was derived from my experience and in the course of my duties with the Rand Water Board and Stewart,Sviridov and Oliver, Consulting Engineers. The extensive knowledge of Engineers in these organizations may therefore be reflected herein although I am solely to blame for any inaccuracies or misconceptions.
I am grateful to my wife Lesley, who, in addition to looking after the twins during many a lost weekend, assiduously typed the first draft of this book.
David SteDhenson.
viii
CONTENTS
1
ECONOMIC PLANNING, 1 Introduction, 1 Pipeline economics, 2 Basics of economics, 8 Methods of analysis, 10 Uncertainty in forecasts, 10 Balancing storage, 13
2
HYDRAULICS, 15 The fundamental equations of fluid flow, 15 Flowhead l o s s relationships, 16 Conventional flow formulae, 16 Rational flow formulae, 18 Minor losses, 2 4
3
PIPELINE SYSTEM ANALYSIS AND DESIGN, 27 Network analysis, 27 Equivalent pipes for pipes in series or parallel, 27 The loop flow correction method, 2 8 The node head correction method, 29 Alternative methods of analysis, 30 Network analysis by linear theory, 32 Optimization of pipeline systems, 33 Dynamic programming for optimizing compound pipes, 34 Transportation programming for least cost allocation of resources, Linear programming for design of least cost open networks, 40 Steepest path ascent technique for extending networks, 44 Design of looped networks, 47
4
WATER HAMMER AND SURGE, 5 3 Rigid water column surge theory, 53 Mechanics of water hammer, 5 4 Elastic water hammer theory, 5 8 Method of analysis, 6 0 Effect of friction, 62 Protection of pumping lines, 6 4 Pump inertia, 66 Pump bypass reflux valves, 6 9 Surge tanks, 7 0 Discharge tanks, 71 Air vessels, 77 Inline reflux valves, 82 Choice of protective device, 8 4
5
AIR IN PIPELINES, 88 Introduction, 8 8 Problems of air entrainment, 88 Air intake at pump sumps, 90 Air absorption at free surfaces, 92 Hydraulic removal of air, 92 Hydraulic jumps, 94 Free falls, 95 Air valves, 95 Head losses in pipelines, 98 Water hammer. 99
37
ix 6
EXTERNAL LOADS, 101 Soil loads, 101 Trench conditions, 101 Embankment conditions, 105 Superimposed loads, 108 Traffic loads, 109 Stress caused by point loads, 109 Line loads, 110 Uniformly loaded areas, 110 Effect of rigid pavements, 1 1 1
7
CONCRETE PIPES, 1 14 The effect of bedding, 114 Prestressed concrete pipes, 116 Circumferential prestressing, 117 Circumferential prestress after losses, 118 Circumferential stress under field pressure, 119 Longitudinal prestressing, 120 Longitudinal stresses after losses, 122 Properties of steel and concrete, 123
8
STEEL AND FLEXIBLE PIPE, 128 Internal pressures, 128 Tension rings to resist internal pressure, 129 Deformation of circular pipes under external load, 132 Effect of lateral support , 134 Stresses due to circumferential bending, 136 More general deflection equations, 138 Stiffening rings t o resist buckling with no side support, 139
9
SECONDARY STRESSES, 145 Stresses at branches, 145 Crotch plates, 145 Internal bracing, 145 Stresses at bends, 156 The pipe as a beam, 157 Longitudinal bending, 157 Pipe stress at saddles, 158 Ring girders, 159 Temperature stresses, 159
10
PIPES, FITTINGS AND APPURTENANCES, 162 Pipe materials, 162 Steel pipe, 162 Cast iron pipe, 162 Asbestos cement pipe, 163 Concrete pipe, 163 Plastic pipe, 163 Line valves, 164 Sluice valves, 165 Butterfly valves, 165 Globe valves, 166 Needle and control valves, 167 Spherical valves, 168 Reflux valves, 169 Air valves, 168 Air vent valves, 169 Air release valves, 170 Thrust blocks, 171 Flow measurement, 175 Venturi meters, 175
Nozzles, 176 Orifices, 176 Bend meters, 177 Mechanical meters, 177 Electromagnetic induction, 178 Mass and volume measurement, 178 Telemetry, 178 11
12
LAYING AND PROTECTION, 181 Selecting a route, 181 Laying and trenching, 181 Thrust b o r e s , 183 Pipe bridges, 185 Underwater pipelines, 185 Joints and flanges, 186 Coatings, 190 Linings, 191 Cathodic protection, 192 Galvanic corrosion, 192 Stray current electrolysis, Thermal insulation, 196
195
PUMPING INSTALLATIONS, 201 Influence of pumps on pipeline design, 201 Types of pumps, 201 Positive displacement types, 201 Centrifugal pumps, 202 Terms and definitions, 203 Impeller dynamics, 206 Pump characteristics curves, 208 Motors, 2 1 0 Pump stations, 211 GENEFW. REFERENCES AND STANDARDS, 214 APPENDIX, 221 Symbols for pipe fittings, 226 Properties of pipe shapes, 2 2 3 Prouerties of water, 2 2 4 Properties of pipe materials, 225 AUTHOR INDEX, 227 SUBJECT INDEX, 2 2 9
CHAPTER 1
ECONOM I C
PLANNING
INTRODUCTION Pipes have been used for many centuries for transporting fluids. The Chinese first used bamboo pipes thousands of years ago, and lead pipes were unearthed at Pompeii.
In later centuries woodstave pipes were used in England.
It was only
with the advent of cast iron, however, that pressure pipelines were manufactured. Cast iron was used extensively in the 19th Century and is still used.
Steel pipes
were first introduced towards the end of the last century, facilitating construction of small and large bore pipelines.
The increasing use of high grade steels and
large rolling mills has enabled pipelines with diameters over 3 metres and working pressure over 10 Newtons per square millimetre to be manufactured.
Welding tech
niques have been perfected enabling longitudinally and circumferentially welded or spiral welded pipes to be manufactured. Pipelines are now also made inreinforced concrete, prestressed concrete, asbestos cement, plastics and claywares, to suit varying conditions.
Reliable flow formulae became available for the design of
pipelines this century, thereby also promoting the use of pipes. Prior to this century water and sewage were practically the only fluids transported by pipeline.
Nowadays pipelines are the most common means for transporting
gases and o i l s over long distances.
Liquid chemicals and solids in slurry form
or i n containers are also being pumped through pipelines on ever increasing scales. There are now over two million kilometres of pipelines in service throughout the world.
The global expenditure on pipelines in 1974 was probably over €5 000 million.
There are many advantages of pipeline transport compared with other methods such as road, rail, waterway and air:
(1)
Pipelines are often the most economic form of transport (considering either capital costs, running costs or overall costs).
(2)
Pipelining costs are not very susceptible to fluctuations in prices, since the major cost is the capital outlay and subsequent operating costs are relatively small.
(3)
Operations are not susceptible to labour disputes as little attendance is required. Many modern systems operate automatically.
Being hidden beneath the ground a pipeline will not mar the natural environment. A buried pipeline is reasonably secure against sabotage A pipeline is independent of external influences such as traffic congestion
and the weather. There is normally no problem of returning empty containers to the source.
It is relatively easy to increase the capacity of a pipeline by installing a booster pump. A buried pipeline will not disturb surface traffic and services.
Wayleaves for pipelines are usually easier to obtain than for roads and railways. The accident rate per ton  km is considerably lower than for other forms of transport. A pipeline can cross rugged terrain difficult for vehicles to cross.
are of course disadvantages associated with pipeline sytems:The initial capital expenditure is often large,
so
if there is any un
certainty in the demand some degree of speculation may be necessary. There is often a high cost involved in filling a pipeline (especially long fuel lines). Pipelines cannot be used for more than one material at a time (although there are multiproduct pipelines operating on batch bases). There are operating problems associated with the pumping of solids, such as blockages on stoppage. It is often difficult to locate leaks or blockages.
PIPELINE ECONOMICS The main cost of a pipeline system is usually that of the pipeline itself.
The pipeline cost is in fact practically the only cost for gravity systems
but as the adverse head increases s o the power and pumping station costs increase. Table 1.1 indicates some relative costs for typical installed pipelines. With the economic instability and rates of inflation prevailing at the time of writing pipeline costs may increase by 20% or more per year, and relative costs for different materials will vary.
In particular the cost of petro
chemical materials such as PVC may increase faster than those of concrete for instance, so these figures should be inspected with caution.
3
TABLE 1.1
RELATIVE PIPELINE COSTS Bore mm 150
450
6
23

Asbestos cement
7
23

Reinforced concrete

23
Prestressed concrete

33
Mild steel
10
28
100  180
High tensile steel
11
25
90  120
Cast iron
25
75
Pipe Material PVC
*
1111
1 500
80 90

150

indicates not readily available.
1 unit
=
€/metre in 1974 under average conditions.
The components making up the cost of a pipeline vary widely from situation to situation but for water pipelines in open country and typical conditions are as follows:Supply of pipe

Excavation

55%(may reduce as new materials are developed) 20%(depends on terrain,may reduce as mechanical excavation techniques improve)
Laying and jointing

5%(may increase with labour costs)
Fittings and specials

5%
Coating and wrapping

2%
Structures (valve chambers, anchors)

2%
Water hammer protection

1%
protection, security structures, fences

1%
Engineering and survey costs

5%
Administrative costs

1%
Interest during construction

3%
Land acquisition, access roads, cathodic
Many factors have to be considered in sizing a pipeline : For water pumping mains the flow velocity at the optimum diameter varies from 0.7 m/s to 2 m/s, depending on flow and working pressure.
It is about I m/s for lowpressure heads and
a flow of 100 e / s increasing to 2 m/s f o r a flow of 1 000 e / s and pressure heads at about 400 m of water, and may be even higher for higher pressures.
4
The capacity factor and power cost structures a l s o influence the optimum flow velocity or conversely the diameter for any particular flow. Fig. 1 . 1 illustrates the optimum diameter of water mains for typical conditions. In planning a pipeline system it should be borne in mind that the scale of operation of a pipeline has considerable effect on the unit costs.
By
doubling the diameter of the pipe, other factors such as head remaining constant, the capacity increases sixfold. On the other hand the cost approximately doubles so that the cost per unit delivered decreases to 1/3 of the original.
It is this scale effect which justifies multiproduct lines.
Whether
it is in fact, economical to install a large diameter main at the outset depends on the following factors as well as scale:Rate of growth in demand (it may be uneconomical to operate at low capacity factors during initial years). (Capacity factor is the ratio of actual average discharge to design capacity). Operating factor ( the ratio of average throughput at any time to maximum throughput during the same period), which will depend on the rate of drawoff and can be improved by installing storage at the consumer's end. Reduced power costs due to low friction losses while the pipeline is not operating at full capacity. Certainty of future demands. Varying costs with time (both capital and operating). Rates of interest and capital availability. Physical difficulties in the construction of a second pipeline if required. The optimum design period of pipelines depends on a number of factors, least being the rate of interest on capital loans and the rate of cost inflation, in addition to the rate of growth, scale and certainty of future demands. In waterworks practice it has been found economic to size pipelines for demands up to 10
to
30 years hence.
For large throughput and high growth
rates, technical capabilities may limit the size of the pipeline, so that supplementation may be required within 10 years. Longer planning stages are normally justified for small bores and low pressures. It may not always be economic to lay a uniform bore pipeline.
Where
pressures are high it is economic to reduce the diameter and consequently the wall thickness.
Fig.l.1
Optimum pumping main diameters f o r a particular set of conditions.
6
I n planning a trunk main with progressive decrease in diameter there
may be a number of possible combinations of diameters.
Alternative layouts
should be compared before deciding on the most economic.
Systems analysis
techniques such as linear programming and dynamic programming are ideally suited for such studies. Booster pump stations may be installed along lines instead of pumping to a high pressure head at the input end and maintaining a high pressure along the entire line.
By providing for intermediate booster pumps at the
design stage instead of pumping to a high head at the input end, the pressure heads and consequently the pipe wall thicknesses may be minimized.
There may
be a saving in overall cost, even though additional pumping stations are required.
The booster stations may not be required for some time.
The capacity of the pipeline may often be increased by installing booster pumps at a later stage although it should be realized that this is not always economic.
The friction losses along a pipeline increase approximately with
the square of the flow, consequently power losses increase considerably for higher flows. The diameter of a pumping main to convey a known discharge can be selected by an economic comparison of alternative sizes. The pipeline cost increases with increasing diameter, whereas power cost in overcoming friction reduces correspondingly. On the other hand power costs increase steeply as the pipeline is reduced in diameter. the present
Thus by adding together pipeline and
value of operating costs, one obtains a curve such as Fig. 1.2,
from which the leastcost system can be selected. An example is given later in the chapter.
There will be a higher cost the greater the design discharge
rate. If at some stage later it is desired to increase the throughput capacity of a pumping system, it is convenient to replot data from a diagram such as Fig. 1.2 in the form of Fig. 1.3.
Thus for different possible throughputs,
the cost, now expressed in cents per kilolitre or similar, i s plotted as the ordinate with alternative (real) pipe diameter a parameter. It can be demonstrated that the cost per unit of throughput for any pipeline is a minimum when the pipeline cost (expressed on an annual basis) is twice the annual cost of the power in overcoming friction. Thus cost in cents per cubic metre of water i s C1 (P) + c2 (d) C =
Q
(1.1)
7
TOTAL COST $
P r e s e n t V a l uu e
/ DIAMETER Fig.
1 .2
O p t i m i z a t i o n of diameter of a pumping pipeline.
COST I N CENTS PER m3'
C1 C2
Q1 Fig.1.3
Optimization of throughput for certain diameters.
Discharge Q
8
=
O f o r minimum C
i . e . C2(d)
=
2 C1(P)
(1.2)
P i s power r e q u i r e m e n t , p r o p o r t i o n a l t o wHQ, w i s t h e u n i t w e i g h t o f w a t c r , H i s t h e t o t a l h e a d , s u b s c r i p t s r e f e r s t o s t a t i c and f t o f r i c t i o n ,
'1 i s pumping r a t e , C2(d) i s t h e c o s t of a p i p e l i n e of d i a m e t e r d , C,(P) i s t h e c o s t of power ( a l l c o s t s c o n v e r t e d t o a u n i t t i m e b a s e ) .
(In a s i m i l a r manner i t can be shown t h a t t h e power o u t p u t of a given 2 i a m e t e r p e n s t o c k s u p p l y i n g a h y d r o  e l e c t r i c s t a t i o n i s a maximum i f t h e f r i . t i o n head l o s s i s one t h i r d of t h e t o t a l head a v a i l a b l e ) . Returning t o Fig.
1 . 3 , t h e f o l l o w i n g w i l l be observed:
A t any p a r t i c u l a r t h r o u g h p u t Q, t h e r e i s a c e r t a i n d i a m e t e r a t which
o v e r a l l c o s t s w i l l be a minimum ( i n t h i s c a s e D ) . 2 A t t h i s d i s t a n c e t h e c o s t p e r t o n of t h r o u g h p u t could be reduced f u r t h e r i f throughput w a s increased. t h r o u g h p u t 0,.
C o s t s would be a minimum a t some
Thus a p i p e l i n e ' s optimum throughput i s n o t t h e same a s
t h e t h r o u g h p u t f o r which i t i s t h e optimum d i a m e t e r . I f Q 1 were i n c r e a s e d by an amount Q, Q4
=
Q , + Q,,
s o t h a t t o t a l throughput
i t may b e economic n o t t o i n s t a l l a second p i p e l i n e
( w i t h optimum d i a m e t e r D ) b u t t o i n c r e a s e t h e f l o w through t h e p i p e 3 w i t h d i a m e t e r D 2 , i . e . 04C4 i s l e s s t h a n Q , C l + Q3C3. A t a l a t e r s t a g e when i t i s j u s t i f i e d t o c o n s t r u c t a second p i p e l i n e
t h e t h r o u g h p u t t h r o u g h t h e o v e r l o a d e d l i n e could b e reduced. The power c o s t p e r u n i t of a d d i t i o n a l throughput d e c r e a s e s w i t h i n c r e a s i n g p i p e d i a m e t e r s o t h e c o r r e s p o n d i n g l i k e l i h o o d of i t b e i n g most economic t o i n c r e a s e t h r o u g h p u t t h r o u g h an e x i s t i n g l i n e i n c r e a s e s w i t h s i z e (Ref. 1 . 1 ) . BASICS OF ECONOMICS Economics i s used as a b a s i s f o r comparing a l t e r n a t i v e schemes o r designs.
D i f f e r e n t schemes may have d i f f e r e n t c a s h flows n e c e s s i t a t i n g some
r a t i o n a l form of comparison.
The c r u x of a l l methods of economic comparison
i s t h e d i s c o u n t r a t e which may be i n t h e form o f t h e i n t e r e s t r a t e on l o a n s
9
Rate m3/s
_ _ _
Pumping c a p a c i t :

c u m u l a t i v e demand =JQdt
Reservoir fullcapacity
required
24h
Fig.1.4.
Graphical calculation of reservoir capacity.
or redemption funds. National projects may require a discount rate different from the prevailing interest rate, to reflect a time rate of preference, whereas private organizations will be more interested in the actual cash flows, and consequently use the real borrowing interest rate. The cash flows, i.e. payments and returns, of one scheme may be compared with those of another by bringing them to a common time basis. flows may be discounted to their present value.
Thus all cash
For instance one pound re
ceived next year is the same as €1/1.05 (its present value) this year if it could earn 5% interest if invested this year.
It is usual to meet capital
expenditure from a loan over a definite period at a certain interest rate. Provision is made for repaying the loan by paying into a sinking fund which also collects interest. The annual repayments at the end of each year required to amount to € 1 in n years is L
(l+r)n  1 where r is the interest rate on the payments into the sinking fund.
(1.3)
If the
interest rate on the loan is R, then the total annual payment i s
R (l+r)n + r  R (l+r)n  1
(1.4)
Normally the interest rate on the loan is equal to the interest rate earned by the sinking fund so the annual payment on a loan of El is (1.5)
1 0
Conversely the present value of a payment of € 1 at the end of each year over n years is
The present value of a single amount of € 1 in n years is 1
(l+r)n (1.7) Interest tables are available for determining the annual payments on loans, and the present values of annual payments or returns, for various interest rates and redemption periods (Ref. 1.2). Methods of Analysis Different engineering schemes required to meet the same objectives may be compared economically in a number of ways.
If all payments and incomes
associated with a scheme are discounted to their present value for comparison, the analysis is termed a present value or discounted cash flow analysis.
On
the other hand if annual net incomes of different schemes are compared, this is termed the rate of return method.
The latter is most frequently used by
private organisations where tax returns and profits feature prominently.
In
such cases it is suggested that the assistance of qualified accountants i s obtained.
Present value comparisons are most common for public utilities.
A form of economic analysis popular in the United States is benefit/cost analysis.
An economic benefit is attached to all products of a scheme, for
instance a certain economic value is attached to water supplies, although this is difficult to evaluate in the case of domestic supplies. Those schemes with the highest benefit/cost values are attached highest priority.
Where
schemes are mutually exclusive such as is usually the case with public utilities the scheme with the largest present value of net benefit is adopted. If the total water requirements of a town for instance were fixed, the leastcost supply scheme would be selected for construction. Uncertainty in Forecasts Forecasts of demands, whether they be for water, oil or gas, are invariably clouded with uncertainty and risk.
Strictly a probability analysis
is required for each possible scheme,i.e. the net benefit of any particular scheme will be the sum of the net benefits multiplied by their probability for a number of possible demands.
Berthouex (ref. 1.3)
recommends under
designing by 5 to 10% for pipelines to allow for uncertain forecasts, but his analysis does not account for cost inflation.
1 1
An alternative method of allowing for uncertainty is to adjust the discount or interest rate: increasing the rate will favour a low capital cost scheme, which would be preferable if the future demand were uncertain. Example A consumer requires 300 E / s of water for 5 years then plans to increase his consumption to 600 f?/s for a further 25 years (the economic life of his factory). He draws for 75% of the time every day. Determine the most economic diameter and the number of pipelines required. a public body paying no tax.
The water is supplied by
Power costs a flat 0.5 p/kWhr, which i n 
cludes an allowance for operating and maintainance.
The interest rate on
loans (taken over 20 years) and on a sinking fund is 10% per annum, and the rate of inflation in cost of pipelines, pumps and power is 6% p.a. Pump and pumpstations costs amount to €300 per incremental kW, (including an allowance for standby plant) and pump efficiency is 70%. The effective discount rate may be taken as the interest rate less the rate of inflation,i.e. 4 % p.a., since € 1 this year is worth € 1 x €1.10/1.06
5 € 1 x € 1 . 0 4 next year.
The supply could be made through one large pipeline capable of handling 600 E/s, or two smaller pipelines each delivering 300 l / s , one installed five years after the other.
A comparison of alternative
diameters is made in the table on p. 12 for a single pipeline. Similarly an analysis was made for two pipelines each delivering 300 E / s .
This indicated an optimum diameter of 600 mm for each pipeline
and a total present value for both pipelines of €7 500 per 100 m.
Thus
one pipeline, 800 mm diameter, will be the most economic solution. Note that the analysis is independent of the length of the pipeline, although it was assumed that pressure was such that a continuous lowpressure pipe was all that was required. Water hammer protection costs are assumed incorporated in the pipe cost here.
The analysis is also independent of the
capital loan period, although the results would be sensitive to change in the interest yr inflation rates.
Discount factors were obtained from present
value tables for 4% over 5, 25 and 30 years periods. Uncertainty was not allowed for but would favour the two smaller pipelines. Another interesting point emerged from the analysis: If a 600 mm pipeline was installed initially, due to a high uncertainty of the demand increasing from 300 to 600 E / s , then if the demand did increase, it would be
12
Solution : 1.
Inside Dia.mm
2.
Flow
3.
Sead loss m/100 m
4.
Power l o s s kW/ 100 m
5.
Energy requirements kW hr/yr/ 100 m
=
( 3 . ) ~4 / 7 0
7.
9.
300
600
300
600
0.24
0.03
0.12
0.02
0.07
0.60
4.72
0.26
2.06
0.13
1.03
0.09
0.60
3900
31000
850
6600
590
3900
20
155
8
67
4
33
3
20
90

40

20

10


2430

1060

520

310
( 6 . ) ~4.452
1700 13500
Equiv. capital cost of pumping over 25 yrs. ( 6 . ) ~ 15.622
Present value of pumping cost =
10.
600
0.06
Equiv.Capita1 cost of pumping over 5 years
=
900
800
300
0.55
Annual pumping cost €/I00 m
=
8.
600
0.14
= ( 4 . ) ~8760 x 0.75
6.
700
600 300
Q/s
(8.)/1.170
2070
260
440
900
Cost of pumps etc. f / 1 0 0 m: = ( 4 . ) 180
1410
80
620
40
310
30
180
11.
Present value of pump cost = ( 1 0 . ) / 1 . 1 7 0 for 180 second stage
1200
80
530
40
260
30
150
12.
Pipeline cost €11OOm
x 300
13.
3600
4200
7140
5750
4800
5400
5560"
5850
TOTAL COST € / l o 0 m for 300 600 Q / s 7.+9.+11.+12.
&
(least cost) more economic t o boost the pumping head and pump the total f l o w through the one existing 600 mm diameter pipeline rather than provide a second 600 mm
pipeline.
This is indicated by a comparison of the present value of pumping
through one 600 mm line ( € 7 140/100 m) with the present value of pumping through two 600 mm lines ( € 7 500/100 m).
1 3
BALANCING STORAGE An aspect which deserves close attention in planning a pipeline system is reservoir storage. Demands such as those for domestic and industrial watzr fluctuate with the season, the day of the week and time of day.
Peakday de
mands are sometimes in excess of twice the mean annual demand whereas peak drawoff from reticulation systems may be six times the mean for a day.
It
would be uneconomic to provide pipeline capacity to meet the peak drawoff rates, and balancing reservoirs are normally constructed at the consumer end (at the head of the reticulation system) to meet these peaks.
The storage
capacity required varies inversely with the pipeline capacity. The balancing storage requirement for any known drawoff pattern and pipeline capacity may be determined with a mass flow diagram: Plot cumulativo drawoff over a period versus time, and above this curve plo: a line with slope equal to the discharge capacity of the pipeline. Move this line down till it just touches the mass drawoff period. Then the maximum ordinate between the two lines represents the balancing storage required (see Fig.l.4). An economic comparison is necessary to determine the optimum storage capacity for any particular system (Ref. 1.5).
By adding the cost of res
ervoirs and pipelines and capitalized running costs for different combinations and comparing them, the system with least total cost is selected. It is found that the most economic storage capacity varies from one day's supply based on the mean annual rate for short pipelines to two days's supply for long pipelines (over 60 km).
Slightly more storage may be economic for smallbore
pipelines (less than 450 mm dia.).
In addition a certain amount o f emergency
reserve storage should be provided; up to 12 hours depending upon the availability o f maintenance facilities. REFERENCES
1.1
J.E.White, Economics of large diameter liquid pipelines, Pipe Line News, N.J., June, 1969.
1.2
Instn.of Civil Engs., An Introduction to Engineering Economics, London
1.3
P.M.Berthouex, Accommodating uncertain forecasts, J.Am.Water Works
1962. Assn., 66 (1)
(Jan. 1971) 14.
1.4 J.M.Osborne and L.D.James, Marginal economics applied to pipeline design, Proc. Am.Soc.Civi1 Engs., 99 (TE3) (Aug., 1973) 637.
1 4
1.5
N.Abramov, Methods of reducing power consumption in pumping water, 1nt.Water Supply Assn. Congress, Vienna, 1969.
L I S T OF SYMBOLS C

cost per unit of throughput
D

diameter
n
number of years
r

R

interest rate on loan
Q
throughput interest rate on sinking fund
15
CHAPTER 2
HY D RAIILI CS THE FUNDAMENTAL EQUATIONS OF FLUID FLOW The three most important equations in fluid mechanics are the continuity equation, the momentum equation and the energy equation.
For steady, incom
pressible, onedimensional flow the continuity equation is simply obtained by equating the flow rate at any section to the flow rate at another section along the stream tube.
By 'steady flow' is meant that there is no variation
in velocity at any point with time. 'Onedimensional' flow implies that the flow is along a stream tube and there is no lateral flow across the boundaries of stream tubes.
It also implies that the flow is irrotational.
The momentum equation stems from Newton's basic law of motion and states that the change in momentum flux between two sections equals the sum of the forces on the fluid causing the change. For steady, onedimensional flow this is AFx
=
P Q AVx
where F is the force,
(2.1) p
is the fluid mass density, Q is the volumetric flow
rate, V is velocity and subscript x refers to the 'x' direction. The basic energy equation is derived by equating the work done on an element of fluid by gravitational and pressure forces to the change in energy. Mechanical and heat energy transfer are excluded from the equation.
In most
systems there is energy l o s s due to friction and turbulence and a term is The resulting equation f o r
included in the equation to account for this.
steady flow of incompressible fluids is termed the Bernoulli equation and is conveniently written as :
where V
=
mean velocity at a section
"* 2g g
=
velocity head ( units of length
=
gravitational acceleration
p
=
pressure
p/y
=
pressure head (units of length)
y
=
unit weight of fluid
Z
=
elevation above an arbitrary datum
he
=
head l o s s due to friction or turbulence between sections 1 & 2.
)
16
The sum of the velocity head plus pressure head plus elevation is termed the total head. Strictly the velocity head should be multiplied by a coefficient to account for the variation in velocity across the section of the c0ndu.t. The average value of the coefficient for turbulent flow is 1.06 and for laminar flow it is 2.0.
Flow through a conduit is termed either uniform or
nonuniform depending on whether or not there is a variation in the crosssectional velocity distribution along the conduit. For the Bernoulli equation to apply the flow should be steady, there should be no change in velocity at any point with time. assumed to be onedimensional and irrotational.
1.e.
The flow is
The fluid should be incom
pressible, although the equation may be applied to gases with reservations. The respective heads are illustrated in Fig. 2.1.
For most practical
cases the velocity head is small compared with the other heads, and it may be neglected.
,ENERGY
LINE
I E N T R A N C E LOSS FRICTl3N LOSS CONTRACTION L O S S
F R I C T I O N LOSS
VELOCITY H f k D v /2g PRESSURE H E A D p/ 8 SECTION
1 ELEVAT!ON
Fig. 2.1 Energy heads along a pipeline. FLOWHEAD LOSS RELATIONSHIPS Conventional flow formulae The throughput or capacityof a pipe of fixed dimensions depends on This head is consumed by friction
the total head difference between the ends. and other (minor) losses.
17
The first friction head loss/flow relationships were derived from field observations.
These empirical relationships are still popular in waterworks The head l o s s /
practice although more rational formulae have been developed.
flow formulae established thus are termed conventional formulae and are usually in an exponential form of the type
v
=
K R~ SY or
s
=
K'$/D~
where V is the mean velocity of flow, K and K' are coefficients, R is the hydraulic radius (cross sectional area of flow divided by the wetted perimeter, and for a circular pipe flowing full, equals one quarter of the diameter) and S is the head gradient (in m head l o s s per m length of pipe). Some of the equations more frequently applied are listed below: Basic Equation
'
Hazen Williams
S=K1(V/Cw) * 8 5 / D 1
Manning
S=K (nV)2/D1*33
Chezy
2
s = K ~ ( v / c ~/D )~
Darcy
S
=
units
SI ' 167
XV2/2gD
K
6.84
f.p.s.units
K
3.03
(2.3)
K2= 6.32
K2= 2.86
(2.4)
K
K
4.00
(2.5)
I
=
=
3
13.13
1
=
=
3
Dimensionless
(2.6)
Except for the Darcy formula the above equations are not universal and It should be borne in mind
the form of the equation depends on the units.
that the formulae were derived for normal waterworks practice and take no account of variations in gravity, temperature or type of liquid. for turbulent flow in pipes over 50 m diameter.
They are
The friction coefficients
vary with pipe diameter, type of finish and age of pipe. The conventional formulae are comparatively simple to use as they do not involve fluid viscosity.
They may be solved directly as they do not re
quire an initial estimate of Reynolds number to determine the friction factor (see next section). for flow.
The rational equations cannot be solved directly
Solution of the formulae for velocity, diameter or friction head
gradient is simple with the aid of a slide rule, calculator, computer, nomograph or graphs plotted on loglog paper.
The equations are of particular
use for analysing flows in pipe networks where the flowlhead l o s s equations have to be iteratively solved many times. The moBt popular flow formula in waterworks practice is the HazenWilliams formula. Friction coefficients for use in this equation are tabulated in Table 2.1.
If the formula is to be used frequently, solution with
the aid of a chart is the most efficient way.
Many waterworks organizations
use graphs of head l o s s gradient plotted against flow for various pipe diameter,
18
and various C values.
A s the value of C decreases with age, type of pipe
and properties of water, field tests are desirable for an accurate assesment of C. TABLE 2.1 HAZENWILLIAMS FRICTION COEFFICIENTS C Type of pipe Condition New
25 years
50 years
Badly corroded
o Id
o Id
150
7 40
140
130
150
130
120
I00
galvanized:
150
130
100
60
Cast iron:
130
110
90
50
80
45
PVC: Smooth concrete,AC: Stee1,bitumen lined,
Riveted steel, vitrified, woodstave:
120
For diameters less than 1 000 mm, subtract 0.1 ( 1

Dmm
) C
1 000
Rational flow formulae Although the conventional flow formulae are likely to remain in use for many years, more rational formulae are gradually gaining acceptance amongst engineers.
The new formulae have a sound scientific basis backed by numerous
measurements and they are universally applicable. Any consistent units of measurements may be used and liquids of various viscosities and temperatures conform to the proposed formulae. The rational flow formulae for flow in pipes are similar to those for flow past bodies or over flat plates (Ref. 2.1). on smallbore pipes with artificial roughness.
The original research was Lack of data on roughness
for large pipes has been one deterrent to use of the relationships in waterworks practice. The velocity in a full pipe varies from zero on the boundary to a maximum in the centre.
Shear forces on the walls oppose the flow and a boundary
layer is established with each annulus of fluid imparting a shear force onto an inner neighbouring concentric annulus.
The resistance to relative motion
of the fluid is termed kinematic viscosity, and in turbulent flow it is im
parted by turbulent mixing with transfer of particles of different momentum between one layer and the next. A boundary layer is established at the entrance to a conduit and this
layer gradually expands until it reaches the centre.
Beyond this point the
19
flow becomes uniform.
The length of pipe required for fully established flow
is given by
0.7 Re114 for turbulent flow.
=
The Reynolds number Re
=
(2.7)
VD/v is a dimensionless number incorporating the
fluid viscosity v which is absent in the conventional flow formulae. Flow in a pipe is laminar for low Re (< 2 000) and becomes turbulent for higher Re (normally the case in practice).
The basic head l o s s equation is derived by sett
ing the boundary shear force over a length of pipe equal to the l o s s in pressure multiplied by the area: Tr
DL
=
yhf 11D*/4
d y ... h f= 4V2/2g
L D 2g
vz = A L D 2g (2.8) (the Darcy friction factor), ‘I is the shear stress, D i s where X = 4Tr( v2/2g the pipe diameter and hf is the friction head loss over a length L. X is a function of Re and the relative roughness e/D. For laminar flow, Poiseuille found that A
=
64/Re i.e. A is independent of the relative roughness. Laminar
flow will not occur in normal engineering practice.
The transition zone be
tween laminar and turbulent flow is complex and undefined but is also of little interest in practice. Turbulent flow conditions may occur with either a smooth or a rough boundary.
The equationsfor the friction factor for both conditions are
derived from the general equation for the velocity distribution in a turbulent boundary layer, V J.rlp
5.75 log XI Y where v is the velocity at a distance y from the boundary. =
(2.9)
For a hydro
dynamically smooth boundary there is a laminar sublayer, and Nikuradse found that y 1
v
a
/
m
so
that
TT
J= 5.75
m + 5.5
log y V
(2.10)
The constant 5.5 was found experimentally. Where the boundary is rough the laminar sublayer is destroyed and Nikuradse found that y ’
=
2/30 where e is the boundary roughness. (2.11)
20
Rearranging equations 2.10 and 2.11 and expressing v in terms of the average velocity V by means of the equation Q 1
JI; =
vdA we get
=
2 log Re fi 0.8
(2.12)
(turbulent boundary layer, smooth boundary ) and (2.13) (turbulent boundary layer, rough boundary) Notice that for a smooth boundary,A is independent of the relative roughness e/D and for a rough boundary it is independent of the Reynolds number Re. Colebrook and White combined Equations 2.12 and 2.13 to produce an equation covering both smooth and rough boundaries as well as the transition zone: 1
fi
=
1.14

e 9.3.5 2 log (5 + m)
(2.14)
Their equation reduces to Equ. 2.12 for smooth pipes, and to Equ. 2.13 for rough pipes.
This semiempirical equation yields satisfactory results
for various commercially available pipes.
Nikuradse's original experiments
used sand as artificial boundary roughness.
Natural roughness is evaluated
according to the equivalent sand roughness.
Table 2.2 gives values of e for
various surfaces. TABLE 2.2
ROUGHNESS OF PIPE MATERIALS (from Ref.2.3)
Value of e in mm for new, clean surface unless otherwise stated. Finish:
Smooth
Average
Rough
0
0.003
0.006
Glass, drawn metals Stee1,PVC or AC
0.015
0.03
0.06
Coated steel
0.03
0.06
0.15
Galanized, vitrified clay
0.06
0.15
0.3
Cast iron or cement lined
0.15
0.3
0.6
Spun concrete or wood stave
0.3
0.6
I .5
Riveted steel
1.5
3
6
Foul sewers,tuberculated water mains
Unlined rock, earth
6
15
30
60
150
300
Fortunately A isnot very sensitive to the value of e assumed. e increases linearly with age for water pipes, the proportionality constant depending on local conditions. The various rational formulae f o r A were plotted on a single graph by Moody and this graph is presented a s Fig.2.2.
The kinematic viscosities
of water at various temperatures are listed in the Appendix.
2 1
F I G . 2.2 .Moody resistance diagram for uniform flow in conduits.
22
Unfortunately the Moody diagram is not very amenable to direct solution for any variable for given values of the dependent variables, and a trial and error analysis may be necessary to get the velocity for the Reynolds number if reasonable accuracy is required.
The Hydraulics Research Station at
Wallingford rearranged the variables in the Colebrook  White equation to produce simple explicit flow/head loss graphs (ref.2.3): Equation 2.14 may be arranged in the form
v
=
2
JZgDs
log
(2%+ 2 . 5 ' v 3.7D
ig1
(2.15)
Thus for any fluid at a certain temperature and defined roughness e, may be plotted in terms of V, D and S. 15OC and e
=
a
graph
Fig.2.3 is such a graph for water at
0.06. The Hydraulic Research Station have plotted similar
graphs for various conditions. The graphs are also available for noncircular sections, by replacing D by 4R. Station rewrote the Colebrook
Going a step further, the Hydraulics Research

White equation in terms of dimensionless
parameters proportional to V, R and S, but including factors for viscosity, roughness and gravity. Using this form of the equation they produced a universal resistance diagram in dimensionless parameters.
This graph was a l s o
published with their charts. Diskin (Ref. 2.5) presented a useful comparison of the friction factors from the HazenWilliams and Darcy equations: The Darcy equation may be written as
or
v
=ma
v
=
(2.16)
cZ JSR
(2.17)
which is termed the Chezy equation and the Chezy coefficient is
cz
=
&gF
(2.18)
The HazenWilliams equation may be rewritten for all practical purposes in the following dimensionless form: S
= 515(V/CW)*
(Cw/Re)0.15 /gD (2.19)
By comparing this with the DarcyWeisbach equation (2.16) it may be deduced that cW = 42~4/'(h0*54Re0'08 ) (2.20) The HazenWilliams coefficient Cw is therefore a function of h and
Re and values may be plotted on a Moody diagram (see Fig.2.2).
It will be ob
served from Fig.2.2 that lines for constant HazenWilliams coefficient coincide with the ColebrookWhite lines only in the transition zone.
In the completely
turbulent zone for nonsmooth pipes the coefficient will actually reduce the
2 3
e = 0.06
Fig.2.3
mm
Flow/head l o s s c h a r t ( R e f . 2 . 4 ) . (for water at 15°C)
g r e a t e r t h e rteynolds number i . e . one cannot a s s o c i a t e a c e r t a i n HazenWilliams c o e f f i c i e n t w i t h a p a r t i c u l a r p i p e as it v a r i e s depending on t h e flow rate.
The HazenWilliams e q u a t i o n s h o u l d t h e r e f o r e be used w i t h c a u t i o n
f o r h i g h Reynolds numbers and rough p i p e s .
I t w i l l a l s o be n o t e d t h a t v a l u e s
of Cw above a p p r o x i m a t e l y 155 a r e i m p o s s i b l e t o a t t a i n i n waterworks p r a c t i c e 6 (Re around 10 )
.
24
The Manning equation is widely used for open channel flow and part full pipes.
The equation is
v = K
R2/3sh (2.21)
n
where K is 1.00 in SI units and 1.486 in ft lb units, and R is the hydraulic radius A/P where A is the cross sectional area of flow and P the wetted perimeter.
R is D/4 for a circular pipe, and in general for noncircular
sections, 4R may be substituted for D. VALUES OF MANNING'S 'n'
TABLE 2 . 3
0.010
Smooth glass, plastic Concrete, steel (bitumen lined), galanized
0.011
Cast iron
0.012
Slimy or greasy sewers
0.013
Rivetted steel, vitrified, woodstave
0.015
Rough concrete
0.017

MINOR LOSSES One method of expressing head l o s s through fittings and changes in section is the equivalent length method, often used when the conventional friction l o s s formulae are used.
Modern practice is to express losses through
fittings in terms of the velocity head i.e. he efficient.
=
KVz/2g where K is the loss co
Table 2 . 4 gives typical l o s s coefficients although valve manufac
turers may also provide supplementary data and loss coefficients K which will vary with gate opening.
The velocity V to use is normally the mean through
the full bore of the pipe or fitting. TABLE 2 . 4 Bends _ _
hB
LOSS COEFFICIENTS FOR PIPE FITTINGS = KB
V2/2g
Bend angle
Sharp
r/D
=
1
2
6
30'
0.16
0.07
0.07
0.06
45O
0.32
0.13
0.10
0.08
60"
0.68
0.18
0.12
0.08
goo
1.27
0.22
0.13
0.08
180'
2.2
90'
r
with guide vanes =
0.2
radius of bend to centre of pipe.
A significant reduction in bend l o s s is possible if the radius is flattened
in the plane of the bend.
25
Valves
hv
Type 
=
KvV2/2g
Opening :
1 I4
112
314
24
5.6
1 .o
0.2
120
7.5
1.2
0.3
Sluice Butterfly
Full
Globe
10
1
4
Needle
0.5
0.6
Ref lux
1

2.5
Contractions and expansions i n cross section Contractions: hc
=
Expansions: hc
KcV2>/2g
=
A2/A1
KcVlZ/2g /A2
Wall  wall
0 0.2
angle
0.4
0.6 0 . 8
1.0
0 0.2
0.4
0.6
0.8
1.0 0
7.5O
. I 3 .08
.05
.02
0
15'
.32 .24
.15
.08
.02
0
30'
.78 .45
.27
.13
.03
0
1.0 .64
.36
.17
.04
0
180'
.5
.37
.25
Entrance and exit losses: he
.15 =
.07
0
KeV2/2g
Protruding
Entrance
Exit
0.8
1 .o
Sharp
0.5
1 .o
Bevelled
0.25
0.5
Rounded
0.05
0.2
REFERENCES 2.1
H.Schlichting, Boundary Layer Theory,4th Edn.,McGraw Hill, N.Y.,1960.
2.2
M L.Albertson, J.R. Barton and D.B.Simons, Fluid Mechanics for Engineers, Prentice Hall, N.J., 1960
2.3
Hydraulics Research Station, Charts for the Hydraulic Designs of Channels and pipes, 3rd edn., H.M.S.O., London, 1969
2.4
M.D.Watson, A simplified approach to the solution of pipe flow problems using the ColebrookWhite method. Civil Eng.inS.A. ,21(7) ,July,1979.p169171.
2.5
M.H.Diskin, The limits of applicability of the HazenWilliams formulae, La Houille Blanche, 6 (Nov.,1960).
26
L I S T OF SYMBOLS
A

cross sectional area of flow
c

HazenWilliams friction factor
C'

friction factor
cz
d

depth of water
D

diameter
e

Nikuradse roughness
f

Darcy friction factor (equivalent to A)

force
Fx
Chezy friction factor

gravitational acceleration

head l o s s
hf

friction head l o s s
K

l o s s coefficient
g
P

wetted perimeter
P

pressure
Q
R

hydraulic radius
Re

L
n
length of conduit Manning friction factor
flow rate Reynolds number hydraulic gradient mean velocity across a section velocity at a point distance along conduit distance from boundary elevation specific weight mass density shear stress kinematic viscosity Darcy friction factor  (f in USA)
27
CHAPTER 3
PIPELINE
SYS EM ANALYS S AND DES
GN
NETWORK ANALYSIS The flows through a system of interlinked pipes or networks are controlled by the difference between the pressure heads at the input points and the residual pressure heads at the drawoff points.
A steadystate flow
pattern will be established in a network such that the following two criteria are satisfied:(1)
The net flow towards any junction or node is zero, i.e., inflow must equal outflow, and
(2)
The net head l o s s around any closed loop is zero, i.e., only one head can exist at any point at any time. The line head losses are usually the only significant head losses and
most methods of analysis are based on this assumption. Head loss relationships for pipes are usually assumed to be of the form h =KfQn/Dm where h is the head l o s s , f is the pipe length, Q the flow and D the internal diameter of the pipe. The calculations are simplified if the friction factor K can be assumed the same for all pipes in the network. Equivalent Pipes for Pipes in Series or Parallel It is often useful to know the equivalent pipe which would give the same head l o s s and flow as a number of interconnected pipes in series or parallel.
The equivalent pipe may be used in place of the compound pipes to
perform further flow calculations. The equivalent diameter of a compound pipe composed of sections of different diameters and lengths in series may be calculated by equating the total head l o s s for any flow to the head loss through the equivalent pipe of length equal to the length of compound pipe:K ( C L?)Q"~D,~= IK?Qn/Dm
(m is 5 in the Darcy formula and 4.85 in the HazenWilliams formula). Similarly, the equivalent diameter of a system o f pipes in parallel is derived by equating the total flow through the equivalent pipe 'e' to the sum of the flows through the individual pipes 'i' in parallel:
Now h
=
h.
i.e. KEeQn/Dem
So
=
KEiQin/Dim
cancelling out Q, and bringing D
and E
to the left hand side,
and if each E is the same, (3.2)
The equivalent diameter could also be derived using a flowlhead loss chart. For pipes in parallel, assume a reasonable head loss and read off the flow through each pipe from the chart.
Read off the equivalent diameter
which would give the total flow at the same head loss.
For pipes in series,
assume a reasonable flow and calculate the total head loss with assistance of the chart.
Read off the equivalent pipe diameter which would discharge
the assumed flow with the total head loss across its length. It often speeds network analyses to simplify pipe networks as much as
possible using equivalent diameters for minor pipes in series or parallel Of course the methods of network analysis described below could always be used to analyse flows through compound pipes and this is in fact the preferred method for more complex systems than those discussed above. The Loop Flow Correction Method The loop method and the node method of analysing pipe networks both involve successive approximations speeded by a mathematical technique developed by Hardy Cross (Ref.3.1). The steps in balancing the flows in a network by the loop method are: Draw the pipe network schematically to a clear scale.
Indicate all in
puts, drawoffs, fixed heads and booster pumps (if present). If there is more than one constant head node, connect pairs of constanthead nodes or reservoirs by dummy pipes represented by dashed lines. Assume a diameter and length and calculate the flow corresponding to fixed head loss.
In subsequent flow corrections, omit this pipe.
Imagine the network as a pattern of closed loops in any order.
To speed
convergence of the solution some of the major pipes may be assumed to form large superimposed loops instead of assuming a series of loops side by side. Use only as many loops as are needed to ensure that each pipe is in at least one loop.
29
(4)
S t a r t i n g w i t h any p i p e assume a flow.
Proceed around a loop c o n t a i n i n g
t h e p i p e , c a l c u l a t i n g t h e f l o w i n e a c h p i p e by s u b t r a c t i n g drawoffs and flows t o o t h e r l o o p s a t nodes.
Assume flows t o o t h e r loops i f unknown. It
Proceed t o n e i g h b o u r i n g loops one a t a t i m e , on a s i m i l a r b a s i s .
w i l l be n e c e s s a r y t o make as many assumptions a s t h e r e are l o o p s .
The
more a c c u r a t e e a c h assumption t h e s p e e d i e r w i l l be t h e s o l u t i o n .
(5) C a l c u l a t e t h e head l o s s i n e a c h p i p e u s i n g a formula such as h
=
KeQn/Dm
o r u s e a flow/head l o s s c h a r t ( p r e f e r a b l e i f t h e a n a l y s i s i s t o be done by hand).
(6)
C a l c u l a t e t h e n e t head l o s s around t h e l o o p , i . e . ,
proceeding around t h e
l o o p , add head l o s s e s and s u b t r a c t head g a i n s u n t i l a r r i v i n g a t t h e I f t h e n e t head l o s s around t h e loop i s n o t z e r o , c o r r e c t
starting point.
t h e flows around t h e loop by adding t h e f o l l o w i n g increment i n flow i n t h e same d i r e c t i o n t h a t head l o s s e s were c a l c u l a t e d :
(3.3) This e q u a t i o n i s t h e f i r s t o r d e r approximation t o t h e d i f f e r e n t i a l of t h e head l o s s e q u a t i o n and i s d e r i v e d as follows:Since
h
=
KeQn/Dm
dh
=
Kk?nQn’dQ Dm
=
(hn/Q)dQ
Now t h e t o t a l head l o s s around e a c h loop should be z e r o , i . e . , E(h + dh)
=
0
ChcCdh
=
0
Z h + Z(hn/Q)dQ
=
0
The v a l u e of hn/Q i s always p o s i t i v e e x c e p t f o r dummy p i p e s between c o n s t a n t head r e s e r v o i r s , when i t i s t a k e n as z e r o .
(7)
I f t h e r e i s a b o o s t e r pump i n any l o o p , s u b t r a c t t h e g e n e r a t e d head
(8)
The f l o w around e a c h loop i n t u r n i s c o r r e c t e d t h u s .
(9)
The p r o c e s s i s r e p e a t e d u n t i l t h e head around e a c h loop b a l a n c e s t o a
from Ch
b e f o r e making t h e f l o w c o r r e c t i o n u s i n g t h e above e q u a t i o n s .
s a t i s f a c t o r y amount.
The Node Head C o r r e c t i o n Method With t h e node method, i n s t e a d of assuming i n i t i a l flows around l o o p s , i n i t i a l heads a r e assumed a t e a c h node.
Heads a t nodes a r e c o r r e c t e d by
s u c c e s s i v e approximation i n a s i m i l a r manner t o t h e way flows were c o r r e c t e d f o r t h e loop method.
The s t e p s i n an a n a l y s i s a r e a s f o l l o w s : 
Draw the pipe network schematically to a clear scale.
Indicate all in
puts, drawoffs, fixed heads and booster pumps. Ascribe initial arbitrary heads to each node (except if the head at that node is fixed).
The more accurate the initial assignments, the
speedier will be the convergence of the solution. Calculate the flow in each pipe to any node with a variable head using the formula Q
=
(hDm/K4?)lfn or using a flowfhead l o s s chart.
Calculate the net inflow to the specific node and if this is not zero, correct the head by adding the amount (3.4)
This equation is derived as follows:Since We require But
Q
=
(hDm/KJ?)lfn
dQ
=
Qdhfnh
Z(Q + dQ) ZQ +Z= nh dH = dh
=
0
=
0
so Flow Q and head l o s s are considered positive if towards the node. H is the head at the node.
Inputs (positive) and drawoffs (negative) at the
node should be included in CQ. Correct the head at each variablehead node in similar manner.i.e. repeat steps 3 and 4 for each node. Repeat the procedure (steps 3 to 5) until all flows balance to a sufficient degree of accuracy.
If the head difference between the ends
of a pipe is zero at any stage, omit the pipe from the particular balancing operation. Alternative Methods of Analysis Both the loop method and the node method of balancing flows in networks can be done manually but a computer is preferred for large networks.
If done
manually, calculations should be set out well in tables or even on the pipework layout drawing if there is sufficient space. Fig.3.1 is an example analysed manually by the node method.
There are standard computer programs
available for network analysis, most of which use the loop method. The main advantage of the node method is that more iterations are required than for the loop method to achieve the same convergence, especially
3 1
\
h +30
E
+32,2 417.2
r21.2 +11.2 +4.3
B H I _
20 17.2 34 , 7
:I ::j:”
35.1 4 3 4.4
11.1
0.7
33.4
* 14.9 i15.1
+i6,a 16.2
150mm x h *I0 *7.2
Q
1OOOrn
2OOmmw 2000m
c
c
Q/h
2.3
23 19
2.6
+9.7
22
2.3
+0.7
6
9
0
0
0
70 1.5 + 5 5 8 0 1.5 + 5 4 8 0 1.5 +52,7 8 0 1.5
+45
33.2
0.S
NOTES HEADS I N METRES, FLOWS I N L I T R E S PER SECOND, DIAMETERS I N MILLIMETRES, LENGTHS I N METRES. ARROWS INDICATE P O S I T I V E DIRECTION OF h & Q (ARBITRARY ASSUMPTION). BLACKENED C I R C L E S INDICATE NODES WITH F I X E D HEADS, NUMBERS I N CIRCLES INDICATE ORDER I N WHICH NODES WERE CORRECTED. HEAD LOSSES EVALUATED FROM Fig.2.3. AH
=
1.85 CQ in CQ/h
Fig.3.1 Example of node method of network flow analysis. if the system is very unbalanced to start with. It is normally necessary f o r all pipes to have the same order of head l o s s . There a r e a number of methods for speeding the convergence. These include overcorrection in some cases, or using a second order approximation to the differentials for calculating corrections.
32 The node head correction method is slow to converge on account of the Also
fact that corrections dissipate through the network one pipe at a time.
the head correction equation has an amplification factor (n) applied to the correction which causes overshoot. The loop flow correction method has the disadvantage in data preparation. Flows must be assumed around loops, and drawoffs are defined indirectly in the assumed pipe flows. The added effort in data preparation and interpretation often offsets the quicker convergence. This is low computing costs compared with data assembly.
so
because of relatively
Trial and error design is
also cumbersome if loop flows have to be changed each time a new pipe is added.
Network Analysis
by Linear Theory
The HardyCross methods of network analysis are suited to manual methods of solution b u t suffer drawbacks in the effort required in comparison with computer orientated numerical methods.
The latter involve the simultaneous
solution of sets of equations describing flow 2nd head balance.
Simultaneous
solution has the effect that very few iterations are required to balance a network when compared with the number of iterations for the loop flow correction and node head correction methods.
On the other hand solution of a large
number of simultaneous equations, even if rendered linear, requires a large computer memory and many iterations. NewtonRaphson techniques for successive approximation of nonlinear equations are mathematically sophisticated but the engineering problem becomes subordinate to the mathematics.
Thus Wood and Charles linearized the head loss
equations, improving the linear approximation at each step and establishing equations for head balance around loops. Isaacs and Mills (Ref. 3.2) similarly linearized the head l o s s equation as follows for flow between nodes i and j:
where the term i n the square root sign is assumed a constant for each iteration. If the Darcy friction equation is employed
cij
=
(
71/4)2
2g D5/
XL
Substitute equation 3.5 into the equation for flow balance at each node:
where Q
j
is the flow from node i to node j, ij There is one such equation for each node.
is the drawoff at node j and Q
negative if from j to i.
If each Q is replaced by the linearized eqpression in 3.5, one has a set ij of simultaneous equations (one for each node) which can be solved for H at each node. The procedure is to estimate H at each node initially, then solve for new H's. The procedure is repeated until satisfactory convergence is obtained.
3 3
OPTIMIZATION OF PIPELINES SYSTEMS The previous section described methods for calculating the flows in pipe networks with or without closed loops. For any particular pipe network layout and diameters, the flow pattern corresponding to fixed drawoffs or inputs at various nodes could be calculated. To design a new network to meet certain drawoffs, it would be necessary to compare a number of possibilities. A proposed layout would be analysed and i f corresponding flows were just
sufficient to meet demands and pressures were satisfactory, the layout would be acceptable.
If not, it would be necessary to try alternative diameters for
pipe sizes and analysis o f flows is repeated until a satisfactory solution is at hand. This trial and error process would then be repeated for another possible layout. Each of the final networks so derived would then have to be costed and that network with least cost selected. A technique of determining the leastcost network directly, without recourse to trial and error, would be desirable.
No direct and positive tech
nique is possible for general optimization of networks with closed loops. The problem is that the relationship between pipe diameters, flows, head losses and costs is not linear and most routine mathematical optimization techniques require linear relationships.
There are a number of situations
where mathematical optimization techniques can be used to optimize layouts and these cases are discussed and described below.
The cases are normally
confined to single mains or treelike networks for which the flow in each branch is known.
To optimize a network with closed loops, random search
techniques or successive approximation techniques are needed. Mathematical optimization techniques are also known as systems analysis techniques (which is an incorrect nomenclature as they are design techniques not analysis techniques), or operations research techniques (again a name not really descriptive). be retained here.
The name mathematical optimization techniques will
Such techniques include simulation (or mathematical
modelling) coupled with a selection technique such as steepest path ascent or random searching. The direct optimization methods include dynamic programming, which is useful for optimizing a series of events or things, transportation programming, which is useful for allocating sources to demands, and linear programming, for inequalities (Refs. 3 . 3 and 3 . 4 ) .
Linear programming usually requires the use
of a computer, but there are standard optimization programs available.
3 4
4 One of the simplest optimization techniques, and indeed one which can normally be used without recourse to computers, is dynamic programing.
The
technique is in fact only a systematic way of selecting an optimum program from a series of events and does not involve any mathematics.
The technique
may be used to select the most economic diameters of a compound pipe which may vary in diameter along its length depending on pressures and flows. For instance, consider a trunk main supplying a number o f consumers from a reservoir.
The diameters of the trunk main may be reduced as drawoff takes
place along the line.
The problem is to select the most economic diameter
for each section of pipe.
FIG. 3 . 2 Profile of pipeline optimized by dynamic programing. A simple example demonstrates the use of the technique. Consider the
pipeline in Fig.3.2.
Two consumers draw water from the pipeline, and the
head at each drawoff point is not to drop below 5 m, neither should the hydraulic grade line drop below the pipe profile at any point.
The elevations
of each point and the lengths of each section of pipe are indicated. The cost of pipe is €0.1 per mm diameter per m of pipe. (In this case the cost is assumed to be independent of the pressure head, although it is simple to take account of s u c h a variation).
The analysis will be started at the down
stream end of the pipe (point A ) .
The most economic arrangement will be with
minimum residual head i.e. 5 m, at point A .
The head, H, at point B may be
anything between 13 m and 31 m above the datum, but to simplify the analysis, we will only consider three possible heads with 5 m increments between them at points B and C.
35
The diameter D of the pipe between A and B, corresponding to each of the three allowed heads may be determined from a head l o s s chart such as Fig. 2.3 and is indicated in Table 3.1 (1)
along with the corresponding cost.
We will also consider only three Rossible heads at point C . of possible hydraulic grade lines between B and C is 3 x 3 these is at an adverse gradient so may be disregarded.
=
The number
9, but one of
In Table 3.1 (11) a
set of figures is presented for each possible hydraulic grade line between
B and C.
Thus if HB
= 13 and HC
=
19 then the hydraulic gradient from C to
B is 0.006 and the diameter required for a flow of 110 L'/s is 310 mm (from Fig. 2.3).
The cost of this pipeline would be 0.1 x 310 x 1 000
=
f31 000.
Now to this cost must be added the cost of the pipe between A and B, in this case €60 000 (from Table 3.1 (I)).
For each possible head HC there is one
minimum total cost of pipe between A and C , marked with an asterisk.
It is
this cost and the corresponding diameters only which need be recalled when proceeding to the next section of pipe.
Inthis example, the next section
between C and D is the last and there is only one possible head at D, namely the reservoir level. In Table 3.1 (111) the hydraulic gradients and corresponding diameters and costs for Section C  D are indicated. To the costs of pipe for this section are added the costs of the optimum pipe arrangement up to C.
This
is done for each possible head at C , and the least total cost selected from Table 3.1
(111).
Thus the minimum possible total cost is €151 000 and the
most economic diameters are 260, 310 and 340 mm for Sections A  B , B  C and C

D respectively.
It may be desirable to keep pipes to standard
daimeters in which case the nearest standard diameter could be selected for each section as the calculations proceed or each length could be made up of two sections; one with the next larger standard diameter and one with the next smaller standard diameter, but with the same total head l o s s as the theoretical result. Of course many more sections of pipe could be considered and the accuracy would be increased by considering more possible heads at each section. The cost of the pipes could be varied with pressures. A booster pump station could be considered at any point, in which case its cost and capitalized power cost should be added in the tables. A computer may prove useful if many possibilities are to be considered, and there are standard dynamic programming programs available. It will be seen that the technique of dynamic programming reduces the number of possibilities to be considered by selecting the least  cost arrangement at each step. Refs. 3.5 and 3.6 describe applications of the technique to similar and other problems.
36
TABLE 3.1
DYNAMIC PROGRAMMING O P T I M I Z A T I O N OF A COMPOUND P I P E
I
HEAD HYDR. AT B GRAD.
DIA.
COST
mm
COST
H~
h
13
.004
300
60000
18
.0065
260
52000
~
D
~~

~
__.__
23
.009
250
50000
I
I1
111
1 1 C'
19
I
hDC lDDCI COST E .006
I
I I 1 29
.OOl
310
1
62000
83000 15100O"
4301 86000 79000 165000
1
37
Transportation programming for leastcost allocation of resources Transportation programming is another technique which normally does not require the use of a computer. The technique is of use primarily for allocating the yield of a number of sourees to a number of consumers such that a leastcost system is achieved.
The cost of delivering the resource
along each route should be linearly proportional to the throughput along that route and for this reason the technique is probably of no u s e in selecting the optimum pipe sizes.
It is of use, however, inselecting a leastcost
pumping pattern through an existing pipe distribution system, provided the friction head is small in comparison with static head, or for obtaining a planning guide before demands are accurately known.
CONSUMER REOUIREMENT
M 10 L / S
SOURCE
N 1 5 L/S
. A
B
12 L / S
YIELD
2 0 LIs
FIG. 3.3 Leastcost allocation pattern for transportation programming example. An example serves to illustrate the technique.
In this example, there
are two sources of water, A and B, and two consumers, M and N. could deliver 12 and 20 respectively. routes A

M, A
respectively.
4?/s
respectively and M and N require 10 and 15 4?/s
Thus there is a surplus of water.

N, B
M
A and B
and B

N are
The cost of pumping along
5, 7, 6 and
v f 1 000 l i t r e s
3 8
TABLE 3.2
TRANSPORTATION PROGRAMMING OPTIMIZATION OF AN ALLOCATION SYSTEM
(1)
SOURCE YIELD
CONSUMER:
M
REQUIREMENTS:
10
N
SURPLUS
EVALUATION NUMBER: 0
A
12
t
6
20
7
EVALUATION NUMBER:
5
B
2
J
13
7
7
15
10
2
(11) A
12
B
20
10
4
7
2
The data are set out in tabular form for solution in Table 3.2 (I). Each row represents a source and each column a demand.
The unit cost of
delivery along each route is indicated in the top right corner of the corresponding block in the table.
The first step is to make an arbitrary
initial a s s i g n y f resources in such a manner that each yield and demand is satisfied.
Starting with the top left block of the table, the maximum
possible allocation is 10.
This satisfies the demand of column M and the
amount is written in the bottom left corner of block AM.
Proceeding to the
next column, since the first column is completed, the maximum possible allocation in the first row is 2 , which satisfies the yield of row A. the next block to considered is in row B, namely column N.
So
Proceed through
the table making the maximum possible assignment at each stage until all resources are allocated (even if to the slack column).
Thus the next allo
cation is the 13 in the second row, then the 7 in the third column. Once an initial allocation is made the figures are rearranged methodically until a leastcost distribution emerges. To decide which would be the
3 9
most profitable arrangement, assign a relative evaluation number to each row and column as follows:Assign the value 0 to row 1 and work out the other evaluation numbers such that the sum of the row evaluation number and column evaIuation number is equal
the cost coefficient for any occupied block.
to
M is 5 , for column N is 7, for row B is 2, and
so
The value for column
on. Now write the sum of
the row and column evaluation numbers beneath the cost coefficient ofeach unoccupied block.
If this sum is bigger than the cost coefficient of the
block, it would pay to introduce a resource allocation into the block.
This
is not easy to see immediately, but stems from the method of determining each evaluation sum from the cost coefficients of occupied blocks.
The
biggest possible rate of improvement is indicated by the biggest difference between the evaluation sum and the cost coefficient. The biggest and in fact in our case the only, improvement would be to introduce an amount into Block BM.
The maximum amount which can be put in block BM is determined by
drawing a closed loop using occupied blocks as corners (see the dotted circuit in Table 3.2 (I)).
Now for each unit which is added to block BM, one
unit would have to be subtracted from block BN, added to block AN and subtracted from block AM to keep the yields and requirements consistent.
In this
case the maximum allocation to BM is 10, since this would evacuate block AM. The maximum redistribution i.e. 10 is made, and the amount in the block at each corner of the closed loop adjusted by 10 to satisfy yields and requirements.
Only one redistribution of resources should be done at a time.
After making the best new allocation, recalculate the evaluation number and evaluation sums as in Table 3.2
(11).
Allocate resource to the most
profitable block and repeat the redistribution procedure until there is no further possible cost improvement, indicated by the fact that there is no evaluation sum greater than the cost coefficient in any block.
In our example
we arrived at the optimum distribution in two steps, but more complicated patterns involving more sources and consumers may need many more attempts. The example can only serve to introduce the subject of transportation programming.
There are many other conditions which are dealt with in text
books on the subject of mathematical optimization techniques such as Ref. 3 . 3 and 3.4 and this example only serves as an introduction. For instance, if two blocks in the table happened to be evacuated simultaneously, one of the blocks could be allocated a very small quantity denoted by 'el say. Computations then proceed as before and the quantity 'el disregarded at the end.
4 0
Linear Programming for Design of LeastCost Open Networks Linear programming is one of the most powerful optimization techniques. The use of a computer is normally essential for complex systems, although the simple example given here is done by hand.
The technique may only be
used if the relationship between variables is linear, so it is restricted in application.
Linear programming cannot be used for optimizing the design
of pipe networks with closed loops without resort to successive approximations.
It can be used to design trunk mains or treelike networks where the
flow in each branch is known.
Since the relationships between flow, head
loss, diameter and cost are nonlinear, the following technique is used to
render the system linear : For each branch or main pipe, a number of preselected diameters are allowed and the length of each pipe of different diameter i s treated as the variable.
The head losses and costs are linearly
proportional to the respective pipe lengths. The program will indicate that some diameter pipes have zero lengths, thereby in effect eliminating them. Any other type of linear constraint can be treated in the analysis. It may be required to maintain the pressure at certain points in the network above a fixed minimum ( a linear inequality of the greaterthanequaltotype) or within a certain range.
The total length of pipe of a certain diameter may
be restricted because there is insufficient pipe available. The example concerns a trunk main with two drawoff points ( F i g . 3 . 4 ) . The permissible diameters of the first leg are 250 and 200 mm, and of the second leg, 200 and 150 mm.
There are thus four variables, X , X ,X 1 2 3
RESERVOIR
L E N G T H rn FLOW I/s DIAMETER mm UNKNOWN L E N G T H m OPTIMUM L E N G T H m HEAD
LOSS m
500
L 00 1L
LO 250
200
2 00
150
50
x2 L50
x3 0
Xl LOO
012
3.2
0
1.68 TOTAL 5.0
x,
FIG. 3 . 4 Leastcost trunk main by linear programming.
41
and X
4
which are the lengths of pipe of different diameters. This simple
example could be optimized by manual comparison of the costs of all alternatives giving the correct head loss, but linear programming is used here to demonstrate the technique. The head losses per 100 m of pipe and costs per m for the various pipes are indicated below:Diameter
Head loss @ 40 21s
mm
m/100 m
250
0.25
200
0 .7 1
@ 14 . t / s
Cost f100/100 m
m/100 m
5
150
0.1
4
0.42
3
The linear constraints on the sytem are expressed in equation form below and the coefficients of the equations are tabulated in Table 3 . 3 (I). Lengths are expressed in hundred metres. #
Lengths Head L o s s
:
x1
=
x2
x3
0.25X1 + 0.71X2 + 0.1X
:
Objective Function
+
5x1 +
:
x4
=
4
+ 0.42X4
=
5
3X4
=
minimum.
+
3 4x3 +
4x2 +
5
The computations proceed by setting all real variables to zero,
so
it
is necessary to introduce artificial slack variables into each equation to satisfy the equality.
The slack variables are designated a, b and c in
Table 3. 3 (I), and their cost coefficients are set at very high values designated m.
To initiate the solution, the slack variables a, b and c are
assigned the values 5 , 4 and 5 respectively (see the third column of Table 3.3 (I)).
The numbers in any particular line of the main body of the table indicate the amount of the programme variable which would be displaced by introducing one unit of the column variable.
Thus one unit of X , would displace
1 unit of a and 0. 2 5 units of c.
To determine whether it is worthwhile replacing any variable in the programme by any other variable, a number known as the opportunity number is calculated far each column. would increase by ( 5 (1
x m)
If one unit of X

(0 x m)

1
was introduced, then the cost
(0.25 x m)) , which is designated
the opportunity value, i.e. the opportunity value for each column is calcullated by multiplying the entries in that column by the corresponding cost coefficients of the programme variable in the second column and subtracting
42
TABLE 3.3 L I N E A R PROGRAMMING SOLUTION OF P I P E PROBLEM
V C ao rs it a cb l oe e y T
I
~~~~3
Prog. Cost V a r i a b l e C o e f f .Amount _ _ _ _ ~ _ _ _ 5 1 a m 5 b
m
4
C
m
5
I
OPPORTUNITY VALUE:
x4 3
4 1
a
ratio
m 1
1 a25
0.71
5L25m
41.7lm
oo:
>
0.42
41.lm
~
3142m
0
042
0.71
0
0
*
KEY COLUMN
key row
I1
4
5
x2 b
m
4
C
m
145
4 0.46 1+Q46m
0.1 0
4l.lm
3142m 1.71m4
0
0
m
m
111
x2 b
x4
5
4
5
m
0.55
3
3.45
IV
 1.1
x2
x1
x4
45
5
a5
3
4
1.69
 1.69
024 328076m
3.38m82
0
x1
x2 4
x3
1
.0.69
1
2.1 6
0.69
1.52
.216
1 0
0
4
0
1.1069m
L1l.lm
5 4
0.7 6
1.1
x4 3
1
1
0.31
0
a m
m
( N O FLIRTHER IMPROVEMENT POSSIBLE)
m
m
m
m
4 3
the total thus formed from the cost coefficient of the column variable.
The
most profitable variable to introduce would be X2, since it shows the greatest cost reduction per unit (or negative opportunity value). is now designated the key column.
The X2 column
The key column is that which shows the
lowest opportunity value (in the cost minimization case).
Only one variable
may be introduced at a time. To determine the maximum amount of the key column variable which may be introduced, calculate the replacement ratios for each row as follows:Divide the amount of the programme variable for each row by the corresponding number in the key column.
The lowest positive replacement ratio is
selected as that is the maximum amount which could be introduced without violating any of the constraints. The row with the lowest positive replacement ratio is designated the key row and the number at the intersection o f the key column and key row, the key number. After introducing a new variable, the matrix is rearranged (Table 3 . 3 (11)) so that the replacement ratios remain correct.
The programme variable
and its cost coefficient in the key row are replaced by the new variable and its cost coefficient.
The amount column as well as the body of the table
are revised as follows:Each number in the key row is divided by the key number. From each number in a nonkey row, subtract the corresponding number in the key row multiplied by the ratio of the old row number in the key column divided by the key number.
The new tableau is given as Table 3 . 3 (11).
The procedure of studying opportunityvalues and replacemnt ratios and revising the table is repeated until there is no further negative opportunity value. so
In the example Table 3 . 3 (IV) shows all positive opportunity values
the leastcost solution is at hand (indicated by the current programme
variables and their corresponding values). The reader should refer to a standard textbook on linear programming (e.g. Ref. 3 . 3 and 3 . 4 ) for a full description of the technique. There are many other cases which can only be mentioned below:(1)
If the constraints are of the 5 (lessthanorequalto) type and not just equations, slack variables with zero cost coefficients are introduced into the 1.h.s. of each constraint to make them equations. The artificial slack variables with high cost coefficients are then omitted
(2)
.
If the constraints are of the 2 (greaterthanorequalto) type, introduce artificial slack variables with high cost coefficients into
44
the 1.h.s. of the constraint and subtract slack variables with zero cost coefficients from each inequality to make them equations. (3)
If the objective function is to be minimized, the opportunity value with the highest negative value is selected, but if the function is to be maximized, the opportunity value with the highest positive value is selected. Note all variables are assumed to be positive.
(4)
The opportunity values represent shadow values of the corresponding variables i.e. they indicate the value of introducing one unit of that variable into the programme.
(5)
If two replacement ratios are equal, whichever row is selected, the amount of programme variable in the other row will be zero when the matrix is rearranged. Merely assume it to have a very small value and proceed as before.
Steepest path ascent technique for extending networks A 'steepest path ascent technique' can be used for extending pipeline
networks at minimum cost (Ref. 3.7).
The technique is primarily for adding
new pipes to existing networks when demands exceed the capacity of the existing pipe networks. It is usually possible to supply points in a system along various routes or from alternative sources.
It may not be necessary to lay a new
pipeline all the way from the source to the demand, or alternatively the diameter of the new pipe may vary from one section to another. The fact that elevation, and hence pressure head, varies along a pipeline route causes different pipe diameters to be optimal at different sections. On account of the complexity of the optimization technique, use of a computer is essential. Alternative routes along which water could be supplied to the node in question are preselected manually.
The computer program is
used to determine the optimum pipelines and corresponding diameters to meet the specific demands. An informal demonstration that the method yields an optimum design is given with the aid o f diagrams. Pipeline costs increase with increasing diameter and wall thickness. Wall thickness usually increases in proportion to the diameter, so the cost per metre will be a function of the square of the diameter. Now for a given head gradient, the discharge is proportional to the pipe diameter to a power of approximately 2.5.
However, if the pipe is not
4 5
laid along the entire route from the source to the demand, but merely reinforces part of the network, the capacity will be limited by the capacity of the remainder of the network.
Hence it may be deduced that discharge varies
with cost of a new pipe section to a power greater than unity, but is limited by the capacity of the remainder of the pipe network. If more than one proposed new pipe is involved the relationship between discharge and pipe costs is multidimensional. Fig. 3.5 illustrates the relationship between discharge at a particular node and cost of two possible pipes in the network.
FIG. 3.5 Relationship between discharge and cost of two pipes.
FIG. 3.6 Steepest path projected onto C,
 C2 plane.
46
The curves on the C1  C2 plane in Fig. 3.5 are lines of constant discharge Q.
The shortesr path between two Q lines spaced a small distance apart
is a line perpendicular to the Q lines. This is the path with the steepest discharge/cost gradient, and is the one sought. The procedure is therefore to start at the origin and proceed in increments on the C,  C
plane, each 2 increment being perpendicular to the next Q line, until the desired discharge
Q is attained.
To determine the increments in diameter corresponding to increments in cost, the actual increment in cost of each pipe for a step on the C 1  C2 plane Consider the triangle XYZ, enlarged in Fig. 3.6:
has to be calculated.
AC and AC corresponding to increment YW in Q are to be determined. 1 2 Now cos 0 = //(
[email protected] + ( AQ/ aQ )2 acl
%
$i2
ac2
ACl =
YW
cos 0
AQ

aQ ) 2 ac1
(
Similarly AC2
=
AQ/
=
+
aQ
cos20
acl
aQ ac1
aQ
()2
ac2
Yw cos 4
= AQ
3 cos2
I$
ac2
,
In a similar manner
for n possible pipes, it may be proved that
1
AC. =
c
aQ aci
(12
i =t,n
(3.6)
The rate of increase of discharge with respect to the cost Ci of any pipe,
aQ ac;
is determined by analyzing the network with and without a small
increment in diameter D..
The increase in discharge, divided by the increase
in cost of pipe i associated with the increase in diameter, gives the required relationship. The increment in discharge per step, AQ, i s preselected s o that the increase in cost of each pipe is yielded by the above equation
4 7
each step. The corresponding increases in diameter are then calculated from the known diameter/cost relationships.
The diameters of the proposed pipes
are increased in steps until the discharge at the specified node is sufficient. A network analysis should be performed after each step to rebalance the system. It will be observed from Fig. 3.5 that it is unlikely that any local maxima will be reached with the technique, as the discharge/cost curves are generally concave upwards, and have few points of inflection. So
far the technique has been used to supplement the supply to only one
node at a time.
It has been found the pipeline system should be well con
ditioned for satisfactory convergence.
It is usually necessary to initialize
the diameters of proposed pipes at a value greater than zero (say of the order of 1 / 4 of the anticipated
final diameter).
Otherwise the linear approx
imation to the differential equations is unrealistic for the initial steps and false results are yielded. Design of Looped Networks It was explained previously that it is not possible to design a pipe reticulation network with closed loops without recourse to trial and error or successive approximations.
The nonlinear flow/head l o s s relationship,
the fact that flow magnitudes and directions are initially unknown, that pipe diameters should conform to standard sizes and be larger than specified minimum sizes, and that certain minimum pressures are required, all pose problems. There are many approaches to the solution for the leastcost looped network, none of which, it should be noted, overcome all the problems and ensure that a true leastcost solution, and not a local peak in the cost function, is at hand.
The solutions are nevertheless invariably more economic
than a network which is designed by standard methods, and offer a starting system for manipulation by the design engineer. Some techniques proposed for achieving leastcost solutions, together with their limitations, are outlined below.
(i)
Loop/Node Correction Method A method of least cost design, which does not depart radically from the
familiar methods of Hardy Cross analysis, was developed by the author. The optimization procedure is not based on linear or nonlinear programming
48
techniques which are unfamiliar to most engineers.
Instead successive cost
revisions are performed for each node and for each loop in the network using a correction based on the differential o f the cost function determined as

f01lows :
Assume any pipe cost C
=
b a D f
(3.7)
where a and b are constants. Now the diameter, D, can be expressed in terms o f flow Q and head loss
h for any pipe : D = ( K t Qn/h)’/m where K, m and n are constants So
C
=
a(Kt Qn/h)b’mt.
Differentiating, dC = (nb/m) (C/Q) dQ(b/m) (C/h) dh (3.8) i.e. the cost of any pipe can be varied in two ways : by varying flow, Q, keeping the head loss, h, constant, and by varying h while maintaining Q constant. Actually, both factors must be considered in designing a leastcost network.
The fact that the diameters and corresponding costs are
functions of the two independent variables is often overlooked in mathematical optimization models. The complete optimization procedure for a network is therefore as f01lows : Assume a pipe layout and assume any reasonable initial diameter for each pipe. Analyse the network using, say, the Hardy Cross method, to determine flows in each pipe and heads at each node.
Any number o f constant head
reservoirs and drawoffs is permitted. For each loop in turn, calculate the sum of dC/dQ pipe in the loop.
=
(nb/m) C/Q for each
If Q is in the assumed positive direction around the
loop take the positive value o f dC/dQ, otherwise the negative value.
Now if CdC/dQ is positive,i.e. cost increases if flow increases, it would pay to reduce the flow in the positive direction around the loop. Conversely if ZdC/dQ is negative, it would pay to increase the flow around the loop,
Subtract or add an increment in flow around the loop
depending on the sign of ZdC/dQ, and decrease or increase the diameter of each pipe in the loop respectively to keep the head losses constant. The maximum size of increment is that which would reduce any flow to zero, or reduce any pipe diameter to a specified minimum size. An increment slightly smaller than this, say half this value, is preferable. Proceed from loop to loop, repeating this analysis.
It is preferable
to proceed in the order of decreasing absolute value of CdC/dQ, which means the loops should be ranked in order before making the flow corrections.
49
(4)
For each node in turn other than fixedhead reservoirs, calculate the sum of dC/dh
=
(b/m) (C/h) for each pipe connecting the node.
Take the
positive value if the head drops towards the node in question and the negative value if the head drops away from the node. positive it pays toreduce
If ZdC/dh is
the head, H, at the node, and if it is nega
tive, it pays to increase the head.
By increasing the head at the node,
pipes leading to the node will have to be increased in diameter and the pipes leading away reduced in diameter to maintain the flows. Conversely a decrease in head will decrease diameters of pipe leading to the node and increase diameters of pipes leading away.
Determine the maximum
change in head permissible to produce a decrease in cost without altering any flow directions o r reducing pipe diameters to less than specified minimums.
The head of the node should also be maintained above the
specified minimum.
Vary the head correspondingly, or preferably limit
the head change to say half the maximum permissible and calculate the new pipe diameters connecting the node. (5)
Repeat steps 3 and 4 until no further improvement in cost is discernable.
No further network analyses are necessary as once the initial
flow balance of step 2 has been achieved it is not unbalanced. Notice however that once the flow directions have been established they cannot be altered.
It is therefore important that the initially assumed
diameters are realistic and that the corresponding flow pattern is generally correct. The technique will yield nonstandard pipe diameters and these will have to be corrected by assuming the nearest standard pipe size or by letting each pipe comprise two sections, one the next standard size greater and the other the next standard size less than the diameter yielded by the analysis. The corresponding length of each section is calculated from the fact that the total head l o s s must equal that indicated by the analysis. The calculations should be performed by computer as they are lengthy and definitely not as simple as those for a Hardy Cross network analysis. (ii) Flow Correction by Linear Programming In the previous section it was demonstrated that for each pipe dC is linearly proportional to dQ and dH. (this is provided the increments in Q and H are small).
If the objective function is taken as the minimzation of
Z dC for each pipe, the problem may be set up as a linear programming optimization problem.
The linear constraints would be:
50
For each node, CdQ in
=
0 where dQ could be positive or negative, and
HB a specified minimum.
The objective function is CdC
=
minimum, where dC is a linear function
of increments in flows in each pipe and heads at each node.
A standard linear
programming program could be used to select the changes in flow along each pipe and head at each node once an initial network is assumed and analysed. The corresponding diameters could then be calculated.
Unless the increments
in flow and head are confined to very small values, the head losses will be unbalanced after the linear programming optimization and a network analysis will be required.
The linear programming optimization and network analysis
should be iterated until there is not further reduction in the total network cost.
The subprogramme for setting up the linear programming tableau is
complicated and a large computer core storage
is required for reasonably
large networks. The core storage required is proportional to the square of the number of pipes, whereas it is proportional to the linear number of pipes for the Loop/Node Correction Method. Nonstandard diameters are yielded, and the flow directions are not altered once an initial assumption is made. (iii)
NonLinear Programming A s the problem of design of a pipe network is nonlinear a standard or
"canned" nonlinear programming computer program could be used.
Many of
these programs are based on the steepest path ascent technique. A subprogram would be required to formulate the constraints and the objective function.
The constraints could be expressed as linear functions but the
objective function is nonlinear. For each node CQ in For fixedhead nodes
=
H
The constraints are:
0 =
a specified value.
For variable head nodes, H> a specified minimum. For each loop Zh
=
0.
The objective function is CC
=
Ca (Ki? Qn/h)(b/m)
=
minimum.
Nonstandard diameters are yielded and flow directions must be assumed beforehand. A s the technique of nonlinear programming is fairly complicated, it may be difficult to debug the program if errors occur. (iv) Optimum Length Method Kally (Ref. 3.8) proposed a method very similar to the linear programming method o f optimization of treelike networks with known flows.
51
Since the flows are likely to redistribute after optimization of a looped network by this method, Hardy Cross analysis is necessary to balance flows at each node, after which a further optimization is performed, and so on. An initial estimation of diameter i s fed into the program, which calculates flows by Hardy Cross analysis.
For a small change in diameter along
a portion of any pipe, the corresponding head changes at various nodes are calculated.
The relationships between head change and length of enlarged
(or reduced) pipes is assumed linear, and the optimum lengths of each section of new diameter calculated by linear programming.
Diameters may be confined
to standard sizes. (v) Equivalent
Pipe Method
Deb (Ref. 3.9) replaces all pipes in a layout by pipes with a common predefined length and equivalent diameters (i.e. such that head losses remain unaffected). An intial flow pattern is assumed and corrected in steps by adjusting pipe sizes for successive loops.
The total pipe cost for any loop is a mini
mum at some flow extreme i.e. with the flow in some pipes in the loop equal to the specified minumum. A constraint limiting the minimum rate of flow through each pipe may be imposed (for reliability and continuity of supply in case of bursts or blockage closed loops and specified minimum flows in pipes are usually required). A minimum cost is assumed for each loop and a flow correction which will bring the loop cost to the assumed figure is calculated.
If the minimum flow
constraint is violated, a new minimum cost figure is assumed and then the flows corrected accordingly.
If the flows are within permissible limits, a
slightlylower loop cost is assumed, and flows corrected again.
This is re
peated until the cost cannot be reduced any more without violating the constraints. The technique yields nonstandard diameters, and is highly dependent on the initial flow assumptions.
The equations involved tend to obscure the
initial assumptions. REFERENCES
3.1
H. Cross, Analysis of flow in networks of conduits or conductors, University of Illinois Bulletin 286, 1936.
3.2 L.T.Isaacs and K.G.Mills, Linear theory methods for pipe networks analysis. Proc. Am.Soc.Civi1 Engrs., 106 (HY7) (July, 1980).
5 2
3.3
B.Van der Veen, Introduction to the Theory of Operational Research,
3.4
G.B.Dantzig, Linear Programming and Extensions, Princeton University
Cleaver Hume, London, 1967. Press, Princeton, 1963. 3.5
E.Kally, Pipeline planning by dynamic computer programming, J . Am. Water Works Assn., (3) (March, 1969).
3.6
N.Buras and Z.Schweig, Aqueduct route optimization by dynamic pro
3.7
D.Stephenson, Operation research techniques for planning and opera
gramming, Proc.Am.Soc.Civi1 Engrs., 95 (HY5) (Sept. 1969). tion of pipeline systems, Proc.Pipeline Engg.Convn., Brintex Exhibs., London, 1970.
3.8
E.Kally, Automatic planning of the leastcost water distribution network, Water and Water Engg., 75(902) (April, 1971) 148.
3.9
A.K.Deb, Leastcost pipe network derivation, Water and Water Engg., 77 (928) (Jan. 1973) 18.
3.10 C.F.Lam, Discrete gradient optimization of water systems, Proc., Am, Soc.Civil Engrs., 99 (HY6) (June,1973). 3.11 J.R.Appleyard, Discussion on Least Cost design of branched pipe network systems. Proc.Am.Soc.Civi1 Engs., 101, (EE4) (Aug. 1975) 5357.
LIST OF SYMBOLS
diameter
h

H

head
K

constant
k?

flow
c D
Q
cost head l o s s
length
53
CHAPTER 4
WATER HAMMER AND SURGE RIGID WATER COLUMN SURGE THEORY Transient pressures caused by a change of flow rate in conduits are often the cause of bursts.
The pressure fluctuations associated with sudden
flow stoppage can be several hundred metres head. Transients in closed conduits are normally classed into two categories: Slow motion mass oscillation of the fluid is referred to as surge, whereas rapid change in flow accompanied by elastic strain of the fluid and conduit is referred to as water hammer. For slow or small changes in flow rate or pressure the two theories yield the same results. It is normally easier to analyse a system by rigid column theory (where
ever the theory is applicable) than by elastic theory. With rigid column theory the water in the conduit is treated as an incompressible mass.
A
pressure difference applied across the ends of the column produces an instantaneous acceleration.
The basic equation relating the head difference between
the ends of the water column in a uniform bore conduit to the rate of change in velocity is derived from Newton's basic law of motion, and is
where h is the difference in head between the two ends, L is the conduit length, v is the flow velocity, g is gravitational acceleration and t is time
. The equation is useful for calculating the head rise associated with
slow deceleration of a water column. It may be used for calculating the water level variations in a surge shaft following power trip or starting up in a pumping line, or power load changes in a hydroelectric installation fed by a pressure pipeline.
The equation may be solved in steps of A t by computer,
in tabular form or graphically.
The following example demonstrates the
numerical method of solution of the equation:Example A 100 m long penstock with a cross sectional area, A l , of 1 m2 is protected against water hammer by a surge shaft at the turbine, with a cross sectional area, A*, of 2  m z and an unrestricted orifice. The initial velocity in the conduit is 1 m/s and there i s a sudden complete load rejection at the turbine.
Calculate the maximum rise in water
level in the surge shaft neglecting friction.
54
Take At= 1 sec. Then from Equ. 4.1, Av By continuity, Ah t 01 12

A vAt/A2 = lv/2 1 Ah = 0 . 5 ~
=
=
=
ghAt/L
=
9.8h/100
=
0.098h.
0.5~.
AV
h
=
0.098h
V
0.5
0.5
0.049
0.951
0.476
0.976
0.096
0.855 0.717
23
0.428
1.404
0.138
34
0.359
1.763
0.173
0.544
45
0.272
2.035
0.199
0.345
56
0.172
2.207
0.216
0.129
67
0.064
2.271"
0.223
0.094
The maximum rise is 2.27 m, which may be compared with the analytical s o l ution obtained from Equ.4.16, of 2.26m.
The accuracy of the numerical method
could be improved by taking smaller time intervals or taking the mean v and h over the time intervals to calculate Ah and Av respectively.
The method can
readily be extended to include head losses, and is calculatororientated. Another useful application of the rigid water column equation is with water column separation. Following the stopping of a pump at the upstream end of a pumping line, the pressure frequently drops sufficiently to cause vap.0ization at peaks along the line.
In such cases the water column beyond the
vapour pocket will decelerate slowly and rigid column theory is sufficiently accurate for analysis. Equ. 4.1 may be integrated twice with respect to time t to determine the distance the water column will travel before stopping.
If the
pumps stop instantaneously the volume of the vapour pocket behind the water column of length P will be Q
=
Atvoz/2gh
where v is the initial flow velocity. MECHANICS OF WATER HAMMER Water hammer occurs, as the name implies, when a column of water is rapidly decelerated (or accelerated).
The rigid water column theory would
indicate an infinitely large head rise if a valve in a pipeline carrying liquid were slammed shut instantaneously. There is, however, a certain amount of relief under these conditions, due to the elasticity of the fluid and of the pipe itself.
Thus if a valve at the discharge end of a pipeline is shut
the fluid upstream of the gate will pack against the gate, causing a pressure rise.
The pressure will rise sufficiently to stop the liquid in accordance
with the momentum law.
The amount of water stopped per unit time depends on
the amount of water required to replace the volume created by the compression of water and expansion of the pipe.
55
It can thus be shown that the relationship between the head rise Ah, the reduction in velocity Av and the rate of progress of the wave front is Ah
=
cAv/g
(4.2a)
where c is the wave celerity and g is gravitational acceleration. This equation is often referred to as Joukowsky's law.
It can further be proved
from a mass balance that c
=
l / / W i
(4.3)
where w is the unit weight of liquid, K is its bulk modulus, d is the pipe diameter, E is its elastic modulus, y the pipe wall thickness and k a factor which depends on the end fixity of the pipe (normally about 0.9).
c may be
as high as 1370 m / s for a rigid walled tunnel or as low as 850 m/s for thin wall steel pipes.
In the case of plastic pipe, or when free air is present
(see Chapter 5 ) , c may be as low as 200 m/s. The pressure wave caused by the valve closure referred to earlier, thus travels upstream, superimposed on the static head, as illustrated in Fig.4.1(1) When the wave front reaches the open reservoir end, the pressure i n the pipe forces water backwards into the reservoir, so that the velocity now reverses and pressure drops back to static (reservoir) pressure again'.
____
PRESSURE WAVE
r
. I
,.,,,,/I
. INITIAL V =V,
Fig.4.1 Water hammer wave at different stages.
A negative wave (2) thus travels dowmstream from the reservoir.
wave front will in turn reach the closed end. pipeline is v
where v
That
Now the velocity in the entire
was the original velocity.
The negative wave of
head amplitude cv /g below static, travels back up the pipe to the reservoir. Upon reaching the reservoir it sucks water into the pipe, sothat the velocity in the pipe reverts to +v
and the head reverts to static head.
The
sequence of waves will repeat itself indefinitely unless damped by friction.
56

1
0 with
FIG.4.2
friction
Head Fluctuations at valve end. The variation in head at the valve will be as indicated in Fig.4.2.
In the case of pumping lines, the most violent change in flow conditions is normally associated with a pump trip.
The water downstream of the pump
is suddenly decelerated, resulting in a sudden reduction in pressure.
The
negative wave travels towards the discharge end (Fig.4.3), where the velocity reverses and a positive wave returns towards the pumps.
The pressure at the
pump alternates from a pressure drop to a pressure rise.
nuated by friction
FIG.4.3 Water hammer head drop after pump trip. The wave is complicated by line friction, changes in cross section, vaporization, or gradual flow variation.
Thus if there is a negative pressure, the
water will vaporize and air will be drawn in via air valves and from solution. The effect of friction can most readily be seen from Fig.4.4.
5 7
which illustrates a wave at different stages as it travels up the pipeline from a rapidly shut valve.
The pressure heads behind the wave are not quite
horizontal, due to the 'packing' effect causing some flow across the wave front. The effect o f changes in section or branch pipes can be included in one equation. The wave head change Ah' after reaching a junction is related to the original head change Aho by the equation Ah'

2 AhoAl/cI A,/cl
+
A2/c2
+
A3/c3
i 2 Aho / ( 1 + A2/AI + A /A + 3 1
+
......
...)
where A. is the cross sectional area of pipe i and c
X
(4.4a)
F'x
cv,
(4.4b) 1
c2
c3'
"Distance from valve"
9
Fig. 4.4 Head at points along a long pipeline with friction, for instantaneous stoppage.
5 8
When a valve in a gravity main is closed gradually, the effect is analogous to a series of minute waves emanating from the valve. be analysed numerically or graphically.
The system could
The graphical form of analysis is
useful for demonstrating the principles of simultaneous solution of the valve discharge equation and the water hammer equation.
In fact the lines drawn on
a graph are very similar to the socalled characteristic lines adopted in numerical solutions. Fig.4.5 illustrates a graphical analysis of a pipeline with a valve at The valve is closed over a period equal.to 4 L/c and the
the discharge end.
valve discharge characteristics for four different degrees of closure are plotted on the graph.
The line relating friction head loss to pipe velocity
is also indicated. To compute the head at the valve at time L/c after initiating closure one applies the water hammer equation 4.2 between point R and S. Ah
=

AV
(c/g)
Ahf
Thus (4.2b)
where Ahf is the difference in friction head between R and S.
Similarly
applying the water hammer equation from S to R, Ah
=
+(c/g)
Av
+
Ahf
(4.2~)
Computations thus proceed along lines of slope + or  c/g. waves peak and then die out due to friction. at t
=
Ultimately the
The maximum head at point S occurs
4 L/c and is 182 metres according to the computations on Fig.4.5.
ELASTIC WATER HAMMER THEORY The fundamental differential wave equations relating pressure to velocity in a conduit may be derived from consideration of Newton's law of motion and the conservation of mass respectively, and are:
(4.6) The last term in equation 4.5 accounts for friction which is assumed to obey Darcy's equation with a constant friction factor.
The assumption of a
steadystate friction factor for transient conditions is not strictly correct. Tests indicate that head losses during transient flow are higher than those predicted using the friction factor applicable to normal flow conditions. Energy is probably absorbed during flow reversals when the velocity is low and the friction factor consequently relatively high.
59
 201
55 SpVALVE
5s 3 2
SHUT
t'
SI
+At
X
t
CHART
S
I L/c
s o
Fig.4.5 Graphical water hammer analysis for slow valve closure with friction.
60
Methods of Analysis A common method of analysis of pipe systems for water hammer pressures used to be graphically (Ref. 4.1).
Friction was assumed to be concentrated at one
end of the pipe or at a few points along the line, and the water hammer equations were solved simultaneously with the valve or pump characteristics on a graph of h plotted against v, for successive time intervals. This method is now largely replaced by computers. There are charts available for reading off the maximum and minimum heads along a pipeline with or without friction for various rates of linear valve closure (Ref. 4.2). Fig.4.6 is such a chart for the maximum and minimum heads at the downstream valve for no line friction.
The valve area is assumed to
reduce linearly to zero over time T and the valve discharge coefficient is assumed constant.
To use the chart calculate the valve closure parameter cT/L
and valve head loss parameter he/(cvo/g). maximum head parameter h'/(cv h/(cvog)
(dashed lines).
Read off on the vertical axis the
g) (full lines) and minimum head parameter
Multiply by cv /g and the answers are the maximum
and minimum heads respectively above and below static head.
Note that the
chart is for linear reduction in area with time, a condition rarely encountered in practice.
The characteristics of sluice and butterfly valves are such that
most of the flow reduction occurs at the end of the valve stroke, with the result that the water hammer heads are higher than predicted by the chart. A more accurate analysis is therefore necessary for important lines. The most economical method of solution of the water hammer equations for particular systems is by digital computer.
Solution is usually by the method
of characteristics (Ref.4.3. and 4.4) which differs little in principle from the old graphical method.
The differential water hammer equations are expressed
in finite difference form and solved for successive time intervals. The conduit is divided into a number of intervals and At is set equal to Ax/c.
x

The
t grid on which solution takes place is depicted in Fig.4.7. Starting from
known conditions along the pipeline at time t, one proceeds to calculate the head
and velocity at each point along the line at time t + At. By expressing equations 4.5 and 4.6 as total differentials and adding, one gets two simultaneous equations involving dh and dv :
= o
(4.7a)
= o
(4.7b)
Equs. 4.7a and 4.7b may be solved for h' and v' at point p at time t + At in P P terms of known h and v at two other points q and r at time t:
6 1
Valve closure factor
c T/L 1.0

U .c
c)
c
* 0.8 3
0
2 L
0
0.6
>
0
n 0
Y
"In 9
0.4
\ '0
z
c
'Zc'
0.2
I
0
2
f
o0
0.2
0.4
0.6
0.8
1.0
Valve head loss factor h , / y
FIG. 4.6 Maximum and minimum head at downstream valve for linear closure
TIME i
1
t+
FIG.4.7x

t Grid for water hammer analysis by characteristics method.
6 2

b;, v' P
=
h
+ h
~
2
c v  v c X d t vrlvrl r +  4+g 2 2gd
vg + vr
hq
+
2
c


vq[vqI
2
(4.8a)
2
(4.8b)
hr
2d
2
At the terminal points, an additional condition is usually imposed; either h is fixed, or v is a function of a gate opening or pump speed. The correct Equ. 4.7a or 4.7b is solved simultaneously with the known condition to evaluate the new h and v at time t + At.
The computations commence at known
conditions and are terminated when the pressure fluctuations are sufficiently damped by friction. Where a branch pipe s occurs or there is a change in diameter, then Equs. 4.8a and 4.8b should be replaced by Equs. 4.9 and 4.10:
Effect  of Friction Fluid friction damps the water hammer waves as they travel along the conduit.
If there is no exciting influence the waves will gradually die away and
the pressure along the conduit will tend to static pressure. The characteristics method of solution by computer accurately predicts the effect of friction provided there is no discontinuity in the wave.
At a
sharp wave front, it is necessary to resort to some other method of analysis. Fortunately an analytical solution is feasible at the wave front. Ludwig (Ref. 4.5) demonstrated that the amplitude of a water hammer wave travelling back along a line with friction following instantaneous stoppage is indicated by the hyperbolic function (see Fig. 4.4): h
=
cv /g [I

tanh
xxv
2
cv (4.11)
In pumping lines following sudden pump stopping the maximum overpressures at the pump will exceed the pumping head if the friction head is greater than approximately 0.7cv /g.
Fig 4.8 indicates minimum and maximum head en
velopes along pipelines with various friction heads following instantaneous stoppage at the upstream end.
To use the chart, multiply the ordinates by
6 3
cv /g and plot the maximum and minimum head envelopes for the correct friction factor hf/(cvo/g),
above or below static on a pipe profile drawing.
The chart is only valid provided there is no column separation when the negative wave travels up the pipeline i.e..the minimum head envelope should at no point fall below 10 m below the pipeline profile.
It has, however,
been found from experience that themaximum heads with column separation are often similar to those without separation.
FIG. 4 . 8 Maximum and minimum head envelopes following instantaneous pump stopping in pipelines with friction.
6 4
Fig. 4 . 8 may also be used to determine the maximum and minimum heads along pipelines with friction following sudden flow stoppage at the downstream end by a closing valve.
Turn the chart upside down and read off
maximum envelope instead of minimum and viceversa. PROTECTION OF PUMPING LINES The pressure transients following power failure to electric motor driven pumps are usually the most extreme that a pumping system will experience. Nevertheless, the overpressures caused by starting pumps should also be checked.
Pumps with steep head/flow characteristics often induce high over
pressures when the power is switched on.
This is because the flow is small
(or zero) when the pump is switched on
a wave with a head equal to the
closed valve head is generated.
so
By partly closing the pump delivery valves
during starting, the overpressures can be reduced. The overpressures caused by closing line valves or scour valves should also be considered. If the pumps supplying an unprotected pipeline are stopped suddenly, the flow will also stop.
If the pipeline profile is relatively close to the
hydraulic grade line, the sudden deceleration of the water column may cause the pressure to drop t o a value less than atmospheric pressure. value to which
The lowest
pressure could drop is vapour pressure. Vaporization or
even water column separation may thus occur at peaks along the pipeline. When the pressure wave is returned as a positive wave the water columns will rejoin giving rise to water hammer overpressures. Unless some method of water hammer protection is installed, a pumping pipeline system will normally have to be designed for a water hammer overhead equal to cvo/g.
In fact this is often done with highpressure lines
where water hammer heads may be small in comparison with the pumping head.
For
short lines this may be the most economic solution, and even if water hammer protection is installed it may be prudent to check that the ultimate strength of the pipeline is sufficient should the protective device fail.
The philosophy behind the design of most methods of protection against water hammer is similar.
The object in most cases is to reduce the downsurge
in the pipeline caused by stopping the pumps.
The upsurge will then be
correspondingly reduced, or may even be entirely eliminated.
The most common
method of limiting the downsurge i s to feed water into the pipe as soon as the pressure tends to drop.
sU K
HYDRAULIC GRADE LINE
DELIVERY RESERVDIR
TAN)
I
I
=A 0
I 4
I
L
IFIG. 4.9
Pipeline profile illustrating suitable locations for various devices for water hammer protection.
Suitable locations for various protective devices are illustrated in Fig. 4 . 9 .
Most of the systems involve feeding water into the pipe.
Observe
that in all cases the sudden momentum change of the water column beyond the tank is prevented so the elastic water hammer phenomenon is converted to a s l o w motion surge phenomenon.
Part of the original kinetic energy of the
water column is converted into potential energy instead of elastic energy. The water column gradually decelerates under the effect of the difference in heads between the ends.
If it was allowed to decelerate the water column
would gather momentum in the reverse direction and impact against the pump to cause water hammer overpressures.
If, however, the water column is
arrested at its point of maximum potential energy, which coincides with the point of minimum kinetic energy, there will be no sudden change in momentum and consequently no water hammer overpressure. The reverse flow may be stopped by installing a reflux valve or throttling device at the entrance to the discharge tank or air vessel, or in the pipeline.
A small orifice bypass to the
reflux valve would then allow the pressures on either side to gradually equalize. Fortunately charts are available for the design of air vessels and for investigation of the pump inertia effects, s o that a water hammer analysis i s not normally necessary.
Rigid water column theory may be employed for
the analysis of surge tank action, and in some cases, of discharge tanks.
If the pipeline system incorporates inline reflux valves or a pump bypass valve, an elastic water hammer analysis is usually necessary.
The
65
analysis may be done graphically or, if a number of solutions of similar systems are envisaged, a computer program could be developed.
Normally the location,
size and discharge characteristics of a protective device such as a discharge tank have to be determined by trial and error.
The location and size of in
line or bypass reflux valves may similarly have to be determined by trial. In these instances a computer program is usually the most economical method of solution, as a general program could be developed, and by varying the design parameters methodically, an optimum solution arrived at. The following sections describe various methods of reducing water hammer in pumping lines, and offer design aids. Pump Inertia If the rotational inertia of a centrifugal pump and motor continue to rotate the pump for a while after power failure, water hammer pressure transients may be reduced.
The rotating pump, motor and entrained water will
continue to feed water into the potential vacuum on the delivery side, thereby alleviating the sudden deceleration of the water column.
The effect is
most noticeable on lowhead, short pipelines. After the power supply to the motor is cut off, the pump will gradually slow down until it can no longer deliver water against the delivery head existing at the time.
If the delivery head is still higher than the suction
head, it will then force water through the pump in the reverse direction, with the pump still spinning in the forward direction, provided there is no reflux or control valve on the delivery side of the pump.
The pump will
rapidly decelerate and gather momentum in the reverse direction, and will act as a turbine under these conditions.
The reverse speed of the pump will
increase until it reaches runaway speed. Under these conditions there is a rapid deceleration of the reverse flow and water hammer overpressures will result. If there is a reflux valve on the delivery side of the pump, the reverse flow will be arrested, but water hammer overpressures will still occur.
The
pressure changes at the pump following power failure may be calculated graphically (see Ref.4.7) or by computer. The pump speed N after a time increment At is obtained by equating the work done in decelerating the pump to the energy transferred to the water
')*2c[
AM
60
1 gh
h = pressure head at tank, Pipeline profile should be convex upwards Pipeline profile preferably convex downwards.
REFERENCES
4.1 H.R.Lupton, Graphical analysis of pressure surges in pumping systems, J.Inst. Water Engs., 7, 1953.
6 6
4.2
D.Stephenson, Water hammer charts including fluid friction, Proc.
4.3
Am.Soc.Civi1 Engs., 92 (HY5) (Sept., 1966) 7194. V.L.Streeter and C.Lai, Water hammer analysis including fluid fricProc. Am.Soc.Civi1 Engs., 88 (HY3) (May, 1963) 79112.
4.4
V.L.Streeter and E.B.Wylie, Hydraulic Transients, McGrawHill, 1967.
4.5
M.Ludwig and S.P.Johnson, Prediction of surge pressures in long oil transmission lines, Proc.Am.Petroleum Inst., N.Y.,30 (5) (1950)
4.6
D.Stephenson, Water hammer protection of pumping lines, Trans. S.A.
4.7
J.Parmakian, Water Hammer Analysis, Dover Public. Inc., N.Y., 1963.
4.8
H.Kinno and J.F.Kennedy, Water
Instn. Civil Engs., 14 (12) (Dec., 1972). hammer charts for centrifugal pump
systems, Proc.Am.Soc.Civi1 Engs., 91 (HY3) (May, 1965) 247270. 4.9
G.R.Rich, Hydraulic transients, Dover publics. Inc., N.Y., 1963.
4.10 D.Stephenson, Discharge tanks for suppressing water hammer in pumping lines, Proc.Intn1. Conf.on Pressure surges, B.H.R.A, Cranfield, 1972. 4.11 H.Kinno, Water hammer control in centrifugal pump systems, Proc.Am. SOC. Civil Engs., 9 4 (HY3) (May, 1968) 619639. LIST OF SYMBOLS A
pipe cross sectional area
B
air vessel parameter vozAL/(gHoS)
C
water hammer wave celerity
C
throttling parameter Z/H
d
pipeline diameter
E
modulus of elasticity of pipe wall material
F
pump rated efficiency (expressed as a fraction)
f
Darcy friction factor
g h
pressure head at an intermediate section of the pipeline
h'
water hammer head rise measured above the delivery head
gravitational acceleration
hf
friction head l o s s
he
head loss through downstream valve fully open
n
head in pipeline measured above pump suction reservoir level (in case of air vessel design, take H as absolute, i.e. plus atmospheric head)
HO
I
pumping head above suction reservoir level pump inertia parameter MNZ/wALHoZ
87
J

pump parameter FMN2c/180w~vogHo
K

b u l k modulus of water
e

length of an intermediate part of pipeline
L

pipeline length
M

moment of inertia of rotating parts of pump, motor and entrained
N

pump speed in rpm
P

pipeline parameter cv /gH
9

volume of water discharged from discharge tank
s

volume of air initially in air vessel
t

time
T

linear valve closure time
v

water velocity in pipeline
v

initial water velocity in pipeline
w

weight of water per unit volume
x

distance along pipeline from pump
Y

wall thickness of pipe
z

head l o s s through air vessel inlet for pipeline
water (
=
mass x radius of gyration*)
0
velocity = V
A

Darcy friction factor
0
CHAPTER 5
AIF?
IN PIPELINES
IN7XODUCTION
It is recognized that air is present in many water pipelines.
The air
may be absorbed at free surfaces, or entrained in turbulent flow at the entrance to the line. bubbles or pockets.
The air may thus be in solution or in free form in
An air pocket implies a relatively large volume of air,
Likely to accumulate on top of the pipe cross section. travel along the line to peaks.
The pockets may
There they will either remain in equilibrium,
be entrained by the flowing water o r be released through air valves. Air in solution is not likely to present many engineering problems.
It
i s only when the pressure reduces sufficiently to permit dissolved air to
€orm bubbles that problems arise.
The water bulks and head losses increase.
The bubbles may coalesce and rise to the top of the pipe to form large pockets.
Flow conditions then become similar to those in partly full drain
pipes, except that in a pipeline it is likely that the system, including the free air, will be pressurized. Air valves are frequently used to eliminate the air which collects on the top of the pipe.
Air valves of a slightly different design are also
used to release large quantities of air during the filling of the line and to draw air in from the atmosphere during vacuum conditions in the line. PROBLEMS OF AIR ENTRAINMENT Air drawn in through a pump supply can have a number of effects. Minute air bubbles or air in solution can promote cavitation  the formation of vacuous cavities which subsequently rapidly collapse and erode the pump or pipe.
This effect occurs in pump impellers in particular due to the
high peripheral speed which lowers pressures.
Air drawn in in gulps can
cause vibration by causing flow to be unsteady. Air in a rising main, whether in solution or in bubble form, may emerge from solution if ambient pressure or temperature is reduced.
Air in free
form will collect at the top of the pipeline and then run up to higher points.
Here it will either escape through air valves or vent pipes or be
washed along by the velocity of the water past the air pocket in the pipe. The latter will result in a head loss due to the acceleration of the water past the air pocket.
Even if air is in bubble form, dispersed in the water,
the friction head l o s s along the pipeline will increase.
Other problems
89
Trapped a i r c a r r i e d along depending on s l o p e of p i p e
Large q u a n t i t i e s drawn i n intermittently
+To
a.
pump
Low l e v e l sump
I.

A i r entrained by f a l l i n g j e t

To pump
t
A i r enters
intake b.
F r e e f a l l i n t o sump
Surface dimple barely detectable
I
L
A i r drawn
intermittentl y from bottom of v o r t e x into intake c. FIG. 5.1
A i r core extends into intake (fully developed entraining vortex)
Vortex f o r m a t i o n I n l e t arrangements conducive t o a i r entrainment.
90
caused by air present in pipes may include surging, corrosion, reduced pump efficiency, malfunctioning of valves or vibrations. Air may be present in the form of pockets on the top of the water, in bubbles, micro bubbles or in solution.
Bubbles range in size from 1 to 5 mm.
Micro bubbles may be smaller, and their rise velocity is s o slow (e.g. 1 mm bubble, 90 mm/s, 0.1 mm bubble, 4 m/s) that they stay in suspension for a considerable time before rising to the surface.
In fact turbulence may
create an equilibrium concentration profile in the same way that sediment stays in suspension in flowing water.
The rise velocity of small air bubbles is
w = gd2/18v where the kinematic viscosity v of water at 2OoC is 1.1
(5.1) x 106 m*/s, d is
bubble diameter and g is gravitational acceleration. For air bubbles to form readily a nucleus or uneven surface should be present.
Then when conditions are correct, air will rapidly be released
from solution.
The capacity of water to dissolve air depends on the temp
erature and pressure.
At 2OoC water may absorb 2% air by volume measured
at standard atmospheric pressure.
This figure varies from 3 . 2 % at O°C down
to 1.2% at 1 0 0 ' . AIR INTAKE AT PUMP SUMPS The major source of air in pumping lines is from the inlet sump or forebay.
Here the water exposed to the air will absorb air at a rate depend
ing on its temperature, pressure and degree of saturation of the water. Air in free form may also be dragged in to the conduit by the flowing water.
The formation of a vortex or drawdown outside the suction pipe entrance
will entice air into the conduit. The higher the entrance velocity and the greater the tubulence, the more air is likely to be drawn in.
This free
air may later dissolve wholly or partially when pressures increase beyond the pump.
In any case it is carried along the conduit. As pressures again
reduce, it may be released from solution. The configuration of the pump sump has an important bearing on the tendency to draw in air.
Extensive studies by Denny andYoung (Ref. 5 . 1 )
Prosser (Ref. 5.2) and others have indicated how to design sumps to minimize air intake.
Circulation in the sump should be avoided by concentric approaches
and straight inflow. downwards upstream.
The inlet should be bellmouthed and preferably facing Hoods (solid or peforated) above the intake minimize
air intake. The degree of submergence should be as great as possible.
Air entraining
drops or hydraulic jumps are to be avoided. Fig. 5.1 indicates some inlet arrangements to be avoided.
31
a.
Air pocket with subcritical flow past
b.
Air pocket with supercritical flow past
C.
Air pocket in equilibrium position
FIG. 5 . 2 Air pockets in pipelines.
92
AIR ABSORBTION AT FREE SURFACES The rate of diffusion of gas across a liquid interface can be expressed in the form
dM_ dt

AK
(CS

C)
(5.2)
where M is the mass rate of transfer per unit time t, A is the area exposed, K is a liquid film constant, C is the gas concentration and C
is the con
centration at saturation. K is a function of temperature, viscosity and turbulence.
This equation is often written in the form
(5.3) where r is the deficit ratio, C
is the concentration at time 0, and V is
the volume of water per surface area A (V/A
=
depth of water).
HYDRAULIC REMOVAL OF AIR Air trapped in a pipe and allowed to accumulate, will gradually increase the air pocket size.
The water crosssectional area will therefore diminish
and the velocity of the water will increase until at some stage, some or all of the air will be dragged along the line by the water.
Alternatively a
hydraulic jump may form in the pipe entraining air and carrying it away in bubble form, as depicted in Fig. 5.2. The relationship between air pocket volume, washout velocity and pipe diameter has been investigated by a number of workers, and these results are summarized by Wisner et a1 (Ref. 5.3). Kalinske and Bliss (Ref.5.4)
produced the following equation for rate
of water flow Q at which removal commences:
% = gD
0.707 tan 0
(5.4)
where D is the pipe diameter and g is gravitational acceleration and 0 is the pipe slope angle.
The term on the left hand side of Kalinske's equation
is similar to the parameter in the specific momentum and energy diagrams (Figs 5.3 and 5 . 4 ) .
In fact it points to the possibility that the air is
removed if the water depth drops below critical.
Then a hydraulic jump forms
downstream which could entrain air. Wisner et a1 (Ref. 5.3) produced data from which the rise velocity of air pockets may be deduced.
They indicate rise velocity is practically
independent of slope and is a function of Reynolds number vD/v and relative volume of air pocket 4U/nD where U is the air pocket volume (see Fig. 5.5)
FIG. 5.3
Specific momentum function for circular conduits
1.8 1.7
1.6
1.5 1.4 1.3
L
1.2 1.1
1.0
0.9 0.8
0.1 0.6 0.5
0.4 0.3 0.2 0.1
Q2
0
D
FIG. 5 . 4 S p e c i f i c energy f u n c t i o n f o r c i r c u l a r c o n d u i t s .
D
2gDAZ
94
0.6
0.5
0.4
6 \
>
0.3
0.2
103
2
FIG. 5.5
104
105
106
= VD/V
Equilibrium velocity of water for air pockets in a pipe sloping at 1 0 '
to 70'
to the horizontal.
HYDRAULIC JUMPS A hydraulic jump draws i n air in the form of bubbles.
These bubbles
exhibit a surprisingly low tendency to coalesce and remain in free form for a l o n g time.
Air may be absorbed from the bubbles by the water, but
this takes a long time and many of the bubbles rise to the surface before they are absorbed. Kalinske and Robertson (Ref. 5.5) found from model experiments that the rate of air entrainment at a hydraulic jump was given by the equation 0.0066(F1

(5.5) where Qd is the volumetric rate of air entrainment, Q is the water flow Qd/Q
=
1)1*4
rate and F, is the upstream Froude number v was derived
lfi.That relationship
for a jump in a rectangular channel with a free surface down
stream. Experiments by the author indicate a considerably greater air entrainment rate in closed pipes (see Fig. 5.6)
95
F W E FALLS A jet falling free into a pool of water has a similar air intake effect to a hydraulic jump.
Oxygen intake at the base of free falls was studied by
Avery and Novak (Ref. 5.6).
For an initi'al oxygen deficiency of 50%, they
found aeration at a rate of up to 1.6 kg 0 /kWh for low head losses, de2
creasing for higher head losses, e.g. at 1 m head l o s s the aeration efficieency was only 1 kg 0 /kWh.
The kW term represents the energy expended or
lost in the jump or fall.
Assuming 2 1 % oxygen in air, this would indicate
2
aeration rates for air up to 8 kg/kWh. Water flowing into inlets with a free fall e.g. morning glory type spillways, or even siphon spillways will also entrain a considerable amount of air (Ref.5.7).
Gravity pipelines are therefore as much of a problem as
pumping lines.
2
0
FIG. 5.6
Air removal at hydraulic jumps in circular conduits.
AIR VALVES Air accumulating in a pocket in a pipe may be released by air valves. These are normally of the 'small orifice' type (typical orifices are up to 3 nun in diameter).
The valve opens when sufficient air i s accumulated in a
chamber to permit a ball to be released from a seat around the orifice. On the other hand air discharge during filling operations (before
pressurization) and air intake during vacuum conditions in the pipe may be through 'large orifice' air valves, the ball of which is released from the seat around the orifice when pressure inside the pipe is low. Fig.lO.6
96
horizontal scour
k
I
This sectio parallel t o hydrauli gradient constitutes1peak
I
\Horizontal I Datum Section of pipeline running parallel t o hydraulic gradient and constitutes peak
I
gradient only Horizontal Daturn Section of pipeline forming peak with respect t o horizontal and 0150 t o hydraulic gradient and peak with respect t o hydraulic gradient only
NO peak
Horizontal
Datum

Horizontal
Datum
Horizontal Long descending section of pipeline
FIG. 5.7
Long ascending section of pipeline
Position of air v a l v e s a l o n g pipelines.
97
illustrates a double air valve with both a small and a large orifice. The size and spacing of air valves will depend on ambient pressures outside the pipe and permissible pressures inside, the size of the pipe and water flow rates. The equations for the discharge rate through an orifice involve the compressibility of air and are as follows (Ref. 5.8).
If the pressure
beyond the orifice p2 is greater than 0 . 5 3 ~(all ~ pressures absolute i.e. gauge plus atmosphere), then k 1 
I
(5.6) For ~ ~ 1 0 . then 5 3 ~the ~ flow becomes critical and flow rate is independent of p2. Then
k+l k+l
(5.7)
where W is the mass rate of flow of air, C is a discharge coefficient,'a' is the orifice area, m is the mass density of air and k is the adiabatic constant. Normally whether the air valve is for letting air into the pipe or out of it, p2 is less than 0 . 5 3 ~ ~ .Then substituting k C
=
=
1.405 for air and
0.5 into the last equation, it simplifies to Qa
where Q
=
0.34
(5.8)
is the volumetric rate of flow of air (in cubic meters per second
if SI units are employed) measured at the initial pressure, a is the area of the orifice, h is the initial absolute head in metres of water and S is the relative density of air at initial pressure.
Since air density is
proportional to the absolute head,this simplifies further to Qa = 100a where Qais the air volume flow rate at initial pressure
(?lorn
(5.9) absolute) in m3/s
and a is in m2. Thus to release 1 % air from a pipe flowing at a velocity of 1 m/s,
and under low head (10 m absolute being the minimum for no vacuum) then
(5.8) reduces to 0.01 x 1 m/s x A i.e. d
=
=
0.34 d 9 . 8 x 10/1.15 x
0.006D
(5.10)
thus the orifice diameter should be about 1% of the pipe diameter to release 1 % air by volume in the pipe.
98
The theory of flow through an orifice was used by Parmakian (Ref. 5.9) to derive equations for the flow through air valves to fill a cavity formed by parting water columns.
If initial head (absolute) in metres of water is
h l , h2 is the head beyond the orifice, g is gravitational acceleration, C is a discharge coefficient for the valve, d is the orifice diameter, D is the pipe diameter, V is the relative velocity of the water column
each side of the air valve, k is the adiabatic constant for air and S
is
the relative density of air, then for h2 >0.53 h l ,
(5.11) and for h 5 0.53h 2 1 d 
L
D
I
(5.12) For air at 300 m above sea level and air temperatures of 24OC plus 20% humidity
to h,
=
then Sa is 1.15 x
and atmospheric pressure is equivalent
9.97 m of water. k is 1.405.
For air valves C is about 0.5.
HEAD LOSSES IN PIPELINES Air suspended in bubble form or in pockets in flowing water will increase the specific volume.
The mean velocity is consquently higher to
convey a certain volume of water per unit time.
The head l o s s will increase
in accordance with an equation such as the DarcyWeisbach equation (5.13) The velocity v to use will be v where v
W
=
(1 + f)vw
(5.14)
i s the velocity of pure water flowing and f is the volumetric
concentration of free air in the pipe. The head l o s s around a stationary air pocket is due primarily to a l o s s of velocity head.
If the size of air pocket is such that a hydraulic
jump forms, the head l o s s may be evaluated from Fig. 5.4.
If the velocity
is subcritical throughout, it may be assumed that the difference in velocity head is lost. though.
It could be established more closely using momentum principles
39
WATER HAMMER The presence of free air in pipelines can reduce the severity of water hammer considerably.
Fox (Ref 5.10) indicates that the celerity (speed)
of an elastic wave with free air is
(5.15)
For large air contents this reduces to c
=
Jgh/f
(5.16)
where p is the mass density of water, I< is its bulk modulus, D is the pipe diameter, b its thickness, E its modulus of elasticity, p is the absolute pressure and f is the free gas fraction by volume. c is reduced remarkablyfor even relatively low gas contents. Thus 2 % of air at a pressure head of 5 0 m of water reduces the celerity from
about 1100 m/s for a typical pipeline to 160 m/s. The Joukowsky water hammer head is C AV (5.17) g where Av is the change in velocity of flow. There is thus a large reduction in
Ah
Ah.
=
If the air collects at the top of the pipe there is no reason to see
why the same equation cannot apply.
Stephenson (Ref. 5.11)
on the other
hand derived an equation for the celerity of a bore in a partly full pipe. The celerity derived from momentum principles is for small air proportions c
=m
(5.18)
where Ah is the head rise behind the bore. 158 m/s for f
=
0.02 and Ah
=
This indicates a celerity of
5 0 m.
There is a school of thought which favours the installation of air valves in pipelines as a means of reducing water hammer overpressures. The intention is primarily to cushion the impact of approaching columns.
Cal
culations will, however, indicate that an excessively large volume of air is required to produce any significant reduction in head.
The idea stems
from the use of air vessels to alleviate water hammer in pipelines.
It
will be realized that air in air vessels is under high pressure initially and therefore occupies a relatively small volume.
Upon pressure reduction
following a pump trip, the air from an air vessel expands according to the equation puk
=
constant.
where U is the volume of air.
The size of air valves to draw in the
necessary volume of air at low (vacuum) pressures will be found on analysis to be excessive for large diameter pipelines.
100
An unusual problem due possibly to air coming out of solution in a rising main was reported by Glass (Ref. 5.12).
Here a thin stream of air
along the top of the line supposedly collapsed on a pressure rise after a pump trip.
After a number of years the pipe burst along a line along the
soffit. REFERENCES 5.1
D.F.Denny, and G.A.J. Young, The prevention of vortices in intakes. Proc., 7th Con. Int.Ass.Pydr.Res., Lisbon.July 1957
5.2
M.J.Prosser, The Eydraulic Design of Pump Sumps and Intakes, BERA,
5.3
P.Wisner, F.M.Mohsen and N.Kouwen, Removal of air from water
and CIRIA, London
.
48pp, July 1977.
lines by hydraulic means. Proc.,ASCE.101, HY2, 11142, p243257. Feb., 1975 5.4
A.A.Kalinske and P.E.Bliss, Removal of air from pipelines by flowing water, Civil Engineering, ASCE,13, 10. p480.0ct., 1943.
5.5
A.A.Kalinske and J.11.Robertson, Closed conduit flow, Trans.ASCE.
5.6
S.T.Avery and P.Novak, Oxygen transfer at hydraulic structures,
108, 2205, p14531516., 1943. Proc. ASCE,HY11, 14190, p15211540, Nov., 1978. 5.7
D.A.Ervine, The entrainment of air in water. Water Power and Dam
5.8
L.S.Marks, Mechanical Engineers Handbook, 5th Ed., IkGraw Hill, N.Y.,
5.9
J.Parmakian, Air inlet valves f o r steel pipelines. Trans.,ASCE, 115
5.10
J.A.Fox, Yydraulic Analysis of Unsteady Flow in Pipe Networks.
Construction. p2730, Dec., 1976. 2235pp. 1951. 2404. p438444. 1950. Macmillan, London.1977. 5.11
D.Stephenson, Prevention of vapour pockets collapse in a pumping
5.12
IJ.L.Glass, Cavitation of a pump pipeline. Proc.Int.Conf.Pressure
line. Trans., South African Inst.Civi1 Engs. 9, (lo), p255261.0ct.,1967. surges. BHRA, Canterbury. 1980. L I S T OF SYMBOLS
A

b

area wall thickness
C

wave celerity
C

discharge coefficient
a
orifice area
l G l
C
I
cO
gas concentration initial concentration concentration at saturation bubble diameter
D

E

modulus of elasticity
f

volumetric concentration of air
F

Froude number
g

gravitational acceleration
h

head
k

adiabatic constant
k

liquid film constant
K

b u l k modulus
m

mass density of air
cs
d
M
pipe internal diameter
mass rate of transfer or specific momentum
P

Q

water discharge rate
'd
pressure

volumetric rate o f air entrainment
r

gas deficit ratio
S

relative density o r specific gravity
t

time
u

air pocket volume
V

water velocity
v

volume of water per surface area
W

rise velocity of bubbles mass rate of flow of air
' I

0

pipe slope angle
V

kinematic viscosity
P

mass density of water
w Y
depth of flow Darcy friction coefficient
102
CHAPTER 6
EXTERNAL LOADS Low pressure pipes, especially sewers, gravity mains or even large diameter pumping mains should be designed for external loads as well as internal loads.
The vertical soil load acting in combination with vacuum pres
sure inside the pipe could cause the pipe to collapse unless the pipe is adequately supported or stiffened. SOIL LOADS The load transmitted to a pipe from the external surroundings depends on a number of factors:
Rigidity of pipe : The more rigid a pipe is relative to the trench sidefill the more load it will take.
The sidefill tends to settle, thus causing a
large part of the backfill to rest on the pipe.
This occurs with flexible
pipes too to some extent, as a pipe is supported laterally by the fill and will not yield as much as a free standing pipe. Type of trench or fill : Fig. 6.1 illustrates various possible installation conditions for pipes.
The load transmitted to the pipe varies with the width
and depth of trench since friction on the sides of the trench affects the resultant load.
Embankment fills may also transmit different loads to a pipe,
depending on the relative settlement of sidefill and topfill. Marston and Spangler (Ref. 6.1) and Clarke (Ref. 6.2) have developed extensive equations and charts for evaluating soil loads on pipes in various trench and embankment conditions. Although many of their assumptions are subject to question, the fact remains that there is as yet no other theory with which to calculate soil loads on pipes
so
the engineer must use this
theory with discretion. Trench Conditions The soil load transmitted to a rigid pipe in a trench depends on the width and depth of,trench and the soil backfill properties.
For a normal
verticalsided trench (Fig. 6.la), fill at the sides of the pipe will settle more than the pipe, and the sidefill support can be neglected.
On the other
hand the friction of the backfill against the sides of the trench takes some of the load.
The cohesion between trench fill and the sides of the
trench i s neglected. The friction developed is therefore proportional to
103
the coefficient of friction between the fill and the sides of the trench, and the ratio K of the active horizontal pressure to the vertical pressure in the soil.
Equating the vertical upward forces on any horizontal slice in
the trench to the downward forces, Marston evaluated the total load on a pipe at depth H:
9 rr
ALTERNATIVE SIDE ,FORM

_ _ _ _ _ _ /. ._._.__ SETTLEMENT LINE b a NARROW TRENCH
dSHALLDW EMBANKMENT POSITIVE PROJECTlON
e :
WIDE TRENCH SHALLOW E M B A N K M E N T
DEEP EMBANKMENT POSITIVE P R O J E C T I O N
c
f :
W I D E TRENCH DEEP EMBANKMENT
EMBANKMENT N E U T R A L PROJECTION
~
NORMAL DENSITY FILL
g SHALLOW E M B A N K M E N T
NEGATIVE P R O J E C B
,
h.
DEEP EMBANKMENT N E GA T I V E P R O J E CT ICIN
~ U N N E LO R HEAM O R THRUST B O R E
FIG. 6.1
Alternative pipe backfills.
,
EMBANKMENT INDUCED T R E N C H
_
1 0 4
Upward s t r e s s a t d e p t h h + dh : W
+ dW
=
W +yBdh

2K t a n 0
Wdh/B
Solution gives W
=
C yBZ
d where t h e l o a d c o e f f i c i e n t C
d
=
1 e
(2KtanOH/B)
(6.1)
2KtanO
r a t i o of l a t e r a l s o i l pressure t o v e r t i c a l load. f o r active s o i l conditions a n g l e of i n t e r n a l f r i c t i o n of b a c k f i l l a n g l e of f r i c t i o n between b a c k f i l l and s i d e s of t r e n c h h e i g h t of f i l l above p i p e trench width u n i t weight of b a c k f i l l m a t e r i a l KtanO normally r a n g e s from 0.11 f o r s o f t c l a y s to 0.16 f o r s a n d s and c o a r s e c r u s h e d s t o n e . C i s g i v e n i n F i g . 6.2 f o r v a r i o u s v a l u e s f o r KtanO. Note d t h a t f o r deep t r e n c h e s , (H/B g r e a t e r t h a n a p p r o x i m a t e l y l o ) , C approaches d a l i m i t i n g v a l u e of 1 / ( 2 KtanO). T h i s i m p l i e s t h a t t h e s i d e f r i c t i o n i n t h e t r e n c h t a k e s more of t h e l o a d t h e d e e p e r t h e t r e n c h . o r wide t r e n c h e s ( H / B < l ) , C W
=
d
For v e r y s h a l l o w
is approximately equal t o H/B,
yHB
so (6.2)
i . e . most of t h e b a c k f i l l l o a d i s t a k e n by t h e p i p e .
The t r e n c h e q u a t i o n i n 
d i c a t e s t h a t t h e l o a d on a r i g i d p i p e i n c r e a s e s i n d e f i n i t e l y a s t h e t r e n c h width increases.
T h i s i s n o t t h e c a s e and a t some t r e n c h w i d t h t h e e q u a t i o n
no l o n g e r a p p l i e s and embankment c o n d i t i o n s ( s e e n e x t s e c t i o n ) t h e n a p p l y ( s e e F i g . 6.1b and 6 . 1 ~ ) . I t i s n e c e s s a r y t o e v a l u a t e t h e l o a d t r y i n g b o t h t h e t r e n c h c r i t e r i o n (using Fig. 6.2)
and t h e embankment c r i t e r i o n ( u s i n g
F i g . 6 . 3 ) , and t o s e l e c t t h a t l o a d which i s l e a s t . I n t h e c a s e of a 'V' t r e n c h , t h e t r e n c h w i d t h t o u s e i s t h a t a t t h e crown o f t h e p i p e .
I n t h i s c a s e $ i n s t e a d of 0 i n t h e formula f o r Cd i s used
s i n c e t h e s h e a r p l a n e i s i n t h e f i l l n o t a g a i n s t t h e s i d e s of t h e t r e n c h . I f t h e s i d e  f i l l i s w e l l compacted and t h e p i p e i s r e l a t i v e l y f l e x i b l e t h e n t h e l o a d on t h e p i p e depends on t h e p i p e d i a m e t e r and n o t t o t h e same ext e n t on t h e t r e n c h width:
W
=
CdyBD
( f l e x i b l e pipe)
Note t h a t f o r s m a l l H , Cd
w
=
Embankment 
G
(6.3)
H / B and in t h i s c a s e
(6.4)
yHD
Conditions
A p i p e bedded on a f i r m s u r f a c e w i t h a wide embankment f i l l over it i s
s a i d t o be under embankment c o n d i t i o n s .
I f t h e crown of t h e p i p e p r o j e c t s
above t h e o r i g i n a l ground l e v e l it i s a p o s i t i v e p r o j e c t i o n ( F i g . 6 . l d and e).
It i s r e f e r r e d t o a s a complete p o s i t i v e p r o j e c t i o n i f t h e p i p e i s r i g i d
and l i e s c o m p l e t e l y above n a t u r a l ground l e v e l , and t h e embankment i s r e l a t i v e l y shallow.
I f t h e p i p e crown i s below t h e n a t u r a l ground l e v e l one h a s
a n e g a t i v e p r o j e c t i o n o r t r e n c h c o n d i t i o n ( F i g s . 6 . l g and h ) .
For v e r y deep
t r e n c h e s under embanments t h e c o n d i t i o n s may be t h e same a s f o r t h e t r e n c h c o n d i t i o n ( r e f e r r e d t o h e r e as t h e complete t r e n c h c o n d i t i o n ) .
I f t h e crown
i s l e v e l w i t h t h e n a t u r a l ground l e v e l i t i s a n e u t r a l p r o j e c t i o n ( F i g . 6 . l f ) .
LOAD COEFICIENT c,
FIG. 6.2
Load c o e f f i c i e n t Cd f o r t r e n c h c o n d i t i o n s .
106
For the complete positive projection case the fill beside the pipe tends to settle more than directly above the pipe.
The sidefill tends to
drag downwards on the fill directly above the pipe and increase the load on the pipe. W
The resulting load per unit length of pipe is =
CcyD2
where the load coefficient ZK~~II$H/D~
cc
=
2Ktanb
(The line labelled complete projection condition in Fig. 6 . 3 gives the value of C
for Ktan $
=
0.19, which is the minimum adverse friction condition).
For deep embankment, only a certain height of fill above the pipe will settle at a different rate to the sidefill.
This height, He, depends on the product
of the pipe projection ratio u and the settlement ratio s where u
=
s
=
projection of pipe crown above original ground level diameter of pipe settlement of adjacent fill originally at crown level settlement of pipe crown reduction in thickness of adjacent fill below crown level.
Thw settlement of adjacent fill includes the settlement of the original ground level and the compaction of the sidefill.
The settlement of the pipe
crown includes settlement of the bottom of the pipe and vertical deflection of the pipe. The value of s normally varies from 0 . 3 for soft yielding foundation soil to 1.0 for rock or unyielding soil foundation. A common value is 0.7 but 0.5 is recommended if the foundation and sidefill are well compacted. The product s u is used in Fig. 6 . 3 to evaluate the resulting load coefficient for various embankment conditions. The equation for C
It is not necessary to evaluate H
.
for deep embankments and incomplete positive pro
jection conditions is more complicated than for the shallow embankment condition, but it was evaluated by Spangler for Ktan $
=
0.19 (i.e. minimum
adverse friction) condition, and these values are given in Fig. 6 . 3 . Fig. 6 . 3 also gives values of C
for the case when the top of the
pipe settles more than the adjacent backfill (i.e. negative evaluated C
for Ktan+
=
s).
Spangler
0.13 (i.e. minimum favourable friction) for this
case, and the values are indicated in Fig. 6 . 3 , labelled complete and incomplete trench conditions trench conditions.
since the mechanics are similar to those for
107
1c
7
FIG. 6.3
Load coefficient C
for embankment conditions.
(from Spangler, stresses in pressure pipelines and protective casing pipes, Proc.ASCE,ST5,1956,plO54). For the negative projection (or incomplete trench) condition (see Fig. 6.lg and 6.lh) with the pipe in a trench under an embankment, W where C
=
yCn B2
(6.6)
is the same as C
by B in all expressions. u
=
in Fig. 6.3 for negative s u but with D replaced In this case the projection ratio is
depth of pipe crown below natural ground level width of
trench
and the settlement ratio is
s
=
settlement of natural ground levelsettlement of level of top of fill in trench reduction in thickness of s o i l in trench above crown.
The settlement of the level of the top of the fill in the trench may include the settlement of the bottom o f the pipe, deflection of the pipe and compaction of the fill in the trench above the crown of the pipe. from  0.1 for u
=
0.5, to  1.0 for u
=
2.0.
s may range
It is possible to reduce the load on pipes under an embankment by removing a certain amount of compacted fill directly above the pipe and replacing it by lightly compacted fill directly, i.e. inducing a negative projection condition (Fig. 6.1i). to 2.0 for u
=
For this condition s may range from 0.5 for u
=
0.5
2.0.
The load per unit length on a completely flexible pipe under an embankment whether in positive or negative projection condition is
W
(6.7)
= yHD
The load on a rigid pipe in a trench up to its crown level (neutral projection condition) is also
yHD.
Clarke suggests that the load on a tunnel or thrust bore is similar to that on
a
pipe in a trench, with a reduction factor allowing for the cohesion
of the material above.
The asumptions in arriving at this conclusion are not
convincing and further theory would be welcome. Clarke also suggests that uniform surcharges of large extent be treated as embankments and the equivalent depth determined.
This is preferable to
using pure elastic theory, as some load transfer must take place between soil fills at different densities and soil masses which settle differently. The soil density which would result in the maximum load on a pipe is the saturated (but not submerged) density.
The submerged conditions are
normally less severe than for the saturated case, as the water pressure would produce a uniform radial pressure less likely to crack a pipe than a pure vertical load. Example

Negative projection case
The top of a pipe in a 2 m wide trench is 1 m below natural ground level and there is a 7 m high embankment above the trench.
Evaluate
the load on the pipe. Projection ratio u
=
Settlement ratio s
=
H/B
=
812
=
1 
=
0.4
0.5. say.
4.
From Fig. 6.3, load coefficient C = 3.1. n Load per m of pipe = Cn yBZ = 3.1 x 18000 x 22/1000
=
220KN/m.
SUPERIMPOSED LOADS
It is customary to use elastic theory to evaluate the pressures transmitted by surface loads to a buried pipe and to assume a semiinfinite, homogeneous, isotropic, elastic material surrounds the pipe.
The fact that
109
t h e s i d e  f i l l may s e t t l e d i f f e r e n t l y t o t h e p i p e i s i g n o r e d i n t h e t h e o r y and s u i t a b l e f a c t o r s s h o u l d be a p p l i e d t o d e c r e a s e t h e t r a n s m i t t e d load i f the pipe i s f l e x i b l e .
U n f o r t u n a t e l y no s u c h f a c t o r s a r e a v a i l a b l e y e t .
I f t h e l o a d a p p l i e d on t h e s u r f a c e i s of l i m i t e d l a t e r a l e x t e n t t h e n t h e induced p r e s s u r e on any h o r i z o n t a l p l a n e below t h e s u r f a c e d e c r e a s e s with depth.
The d e e p e r
t h e p i p e , t h e wider t h e s p r e a d and c o n s e q u e n t l y t h e
lower t h e maximum p r e s s u r e .
On t h e o t h e r hand t h e t o t a l v e r t i c a l f o r c e must
remain e q u a l t o t h e superimposed
load.
Formulae f o r e v a l u a t i n g t h e t r a n s m i t t e d l o a d s due t o a s u r f a c e p o i n t load, o r a p r e s s u r e of l i m i t e d o r i n f i n t e e x t e n t , a r e g i v e n i n t h e s e c t i o n s below. A s u i t a b l e impact f a c t o r should a l s o be a p p l i e d .
The impact f a c t o r
w i l l v a r y w i t h t h e t y p e of l o a d and d e p t h of p i p e below t h e s u r f a c e , and v a r i e s from 1.3 f o r slow moving v e h i c l e s t o 2.0 f o r f a s t moving v e h i c l e s i f t h e pipe is shallow.
The impact f a c t o r r e d u c e s f o r cover d e p t h g r e a t e r t h a n
t h e p i p e d i a m e t e r and i s o l a t e d p o i n t l o a d s , b u t t h e amount o r r e d u c t i o n i s d e b a t a b l e and it may be s a f e t o adopt t h e f u l l v a l u e of t h e impact f a c t o r u n t i l conclusive fiqures a r e available
T r a f f i c Loads The f o l l o w i n g l o a d s are u s e f u l f o r d e s i g n purposes (Ref. 6.4) (1)
I n f i e l d s : two wheel l o a d s e a c h 30 kN spaced 0.9 m a p a r t .
(2)
Under l i g h t r o a d s : two wheel l o a d s e a c h 72 kN spaced 0.9m a p a r t .
(3)
Under main r o a d s and heavy t r a f f i c (BS 153 t y p e HB l o a d ) : e i g h t wheel l o a d s e a c h 1 1 2 kN comprising two rows 1.8 m a p a r t of f o u r wheels e a c h 0 . 9 m a p a r t l a t e r a l l y , measured from i n n e r a x i s t o i n n e r a x i s .
(4)
A i r p o r t runways : f o u r 210 kN wheel l o a d s spaced a t 1.67 m x 0.66 m.
(5)
Railways (Cooper E72 locomotive) : locomotive and t r a c k uniformly spaced o v e r 2.42 m x 6.1 m a t 0.77 kN/m2.
P i p e l i n e s would n o r m a l l y b e d e s i g n e d f o r f i e l d l o a d i n g and s t r e n g t h e n e d under heavier t r a f f i c .
I f c o n s t r u c t i o n l o a d s are l i k e l y t o exceed d e s i g n l o a d s
e x t r a t o p  f i l l s h o u l d be p l a c e d t o s p r e a d t h e l o a d d u r i n g c o n s t r u c t i o n . St res s caused by P o i n t Loads
Boussinesque's e l a s t i c t h e o r y g i v e s t h e v e r t i c a l stress a t d e p t h H below a p o i n t l o a d and a d i s t a n c e X h o r i z o n t a l l y from t h e p o i n t load P a s
w
=
3P H3
2 T ( H Z + X 2 ) 512.
(6.8)
1 10
The stress at any point due to two or more loads will be the sum of the stresses due to the individual loads.
The maximum stress could occur direct
ly under either load or somewhere between them, and the worst case should be selected by plotting the stress between them. For large pipe diameter, the stress will vary appreciably across the pipe diameter and some average value should be taken, The Concrete Pipe Association (Ref. 6.4) lists the total load in kg/m of pipe for surface wheel loads (l), ( 2 ) and ( 3 ) in the previous section, and various pipe diameters and depths. Line Loads The vertical stress at any depth H below a line load of intensity q per unit length is, by elastic theory, w
=
2q~3
T(X2 + H2)2
(6.9)
Uniformly loaded areas The stress intensity at any depth beneath a loaded area on the surface could be evaluated by assuming the distributed load to comprise a number of point loads, and summating the stress due to each load.
Fortunatley Newmark
has performed the integration of infinitesimal point loads to give the vertical stress under the corner of an uniformlly loaded rectangular area. Newmark's influence coefficients are given in Table 6.1 to evaluate the vertical stress at any point under a surface loaded area, divide the loaded TABLE 6.1
INFLUENCE FACTORS FOR VERTICAL PRESSURE UNDER THE CORNER OF AN UNIFORMLY LOADED RECTANGULAR AREA (from Ref. 6.5)
1 1 1
a r e a i n t o r e c t a n g l e s e a c h w i t h one c o r n e r d i r e c t l y above t h e p o i n t i n q u e s t ion.
C a l c u l a t e t h e r a t i o s L / H and Y / H where L and Y are t h e l e n g t h and
b r e a d t h of t h e r e c t a n g l e and r e a d t h e c o r r e s p o n d i n g i n f l u e n c e f a c t o r from the table.
E v a l u a t e t h e s t r e s s due t o e a c h r e c t a n g l e of load by m u l t i p l y i n g
t h e l o a d i n t e n s i t y by t h e i n f l u e n c e c o e f f i c i e n t f o r t h e p a r t i c u l a r r e c t a n g l e , and summate t h e s t r e s s due t o e a c h such r e c t a n g l e .
I f the point i n question
f a l l s o u t s i d e t h e b o u n d a r i e s of t h e loaded a r e a assume t h e loaded a r e a ext e n d s t o above t h e p o i n t and s u b t r a c t t h e stress due t o t h e imaginary e x t e n s 
ions. S i m i l a r i n f l u e n c e c o e f f i c i e n t s a r e a v a i l a b l e f o r t h e s t r e s s e s under loaded c i r c l e s and s t r i p s , b u t normally any shape can be r e s o l v e d i n t o r e c t a n g l e s o r s m a l l p o i n t l o a d s w i t h o u t much e r r o r .
There a r e a l s o i n f l u e n c e
c h a r t s a v a i l a b l e f o r e v a l u a t i o n of s t r e s s e s under any shape loaded area (Ref. 6.5) The stress a t any d e p t h under an uniform l o a d of v e r y l a r g e e x t e n t i s t h e i n t e n s i t y of t h e s u r f a c e l o a d s i n c e t h e r e i s no l a t e r a l d i s p e r s i o n .
This
could a l s o be deduced from Table 6.1 f o r l a r g e L and Y.
E f f e c t of R i g i d Pavements
A r i g i d pavement above a p i p e has t h e e f f e c t of d i s t r i b u t i n g t h e load l a t e r a l l y and t h e r e f o r e r e d u c i n g t h e s t r e s s i n t e n s i t y due t o a s u r f a c e l o a d . The s t r e s s a t a d e p t h H below a r i g i d pavement of t h i c k n e s s h , i s
w
=
CrP/R2
where R
=
dv
(6.10) =
r a d i u s of s t i f f n e s s
12 (1  v 2 ) k o r 4 000 000 p s i f o r c o n c r e t e )
E
=
modulus of e l a s t i c i t y (28 000 N/mmz
h
=
pavement t h i c k n e s s
W
=
P o i s s o n ' s r a t i o (0.15 f o r c o n c r e t e )
k
=
modulus of subgrade r e a c t i o n , which v a r i e s from 0.028N/mm3 f o r poor s u p p o r t t o 0.14 N/mm3. A t y p i c a l v a l u e f o r good s u p p o r t i s 0.084 N/mm3
R i s normally about t h r e e t i m e s t h e s l a b t h i c k n e s s .
C
i s a function
o f H / R and X / R and i s t a b u l a t e d i n Table 6.2 f o r a s i n g l e p o i n t load P on t h e s u r f a c e .
A c o n c r e t e s l a b normally h a s t h e e f f e c t of about f i v e t i m e s i t s t h i c k n e s s of s o i l f i l l i n a t t e n u a t i n g a p o i n t l o a d .
112
TABLE 6.2
PRESSURE COEFFICIENTS FOR A POINT LOAD ON A PAVEMENT (From Ref.6.3)
W
Pressure on pipe
H
Depth of Top of Pipe Below Pavement
P
Point Load
R X
Radius of Stiffness
30
I I3
04 08 12
101
16
6.1
CrP/R2
Eorizontal Distance from Point Load XiR H     __  ._ R 20 24 28 1 6 04 0 8 12 00 __ . __ __ _ _ _   __ 089 076 062 05 I 043
01 I Oll
032
020 021 022 022
036 03 I
039 035 030 027
030 028 026 023
022 n22 021 019
016 016 016
026 023 020 018
024 021 019 017
02I 019 018
018 016 015 014
014
013 012 010 009
012
105 095 084 072
089 082 074 065
OGR
059
054 046 039 033
047 042
Ob5
061 054
048 047 045 043
032 031 031
037
049 041 036
32 36 40 43
032 027 024 020
030 026 023 020
029
48 52 56 60
018 015 014 012
017 015 013 012
017 014 013 011
016 014 012
015
011
010
64 68 72 76 80
011 010
010 009 008
010
010 009
009 008
008
008
nos
007
007 00 7
007 006
007 006
20 24 28
REFERENCES
=
009 008 007
075
022 019
009 008
008
ooa
007
007 
  __
013 01 I
015
no8
  
01 I
010 009
008 007 007 006 096 .
012 014

_
~
32
40 36 _ _ 
006 004 005 008
n02
Oll Oil 01 I Oil
0n7 OLS
rlni
0110 000
002 005
001 003
005 C'C
008
onb
009
006
01 I Oll 011 010
009 009
007 007 007 007
Oll oin Oil9
on9
007 006
008
on7
008 007 007 007
007 007 006
007
O(l6
00G
006 006 006 005
005 005 005 005
005
005
015
014 013 012
006
008 008
006 006 005
__
ooq
009

006
006
M.G.Spangler, Stresses in pressure pipelines and protective casing pipes, Proc.Am.Soc.Civi1 Engrs., 82 (ST5) (Sept., 1956) ,10541115.
6.2
N.W.B.Clarke, Buried pipelines, MacLaren, London, 1968.
6.3
American Concrete Pipe Assn., Design ManualConcrete Pipe, Arlington, 1970
6.4
Concrete Pipe Assn., Loads on Buried Concrete Pipes, Tech.Bull.No.2,
6.5
P.L.Capper and W.E.Cassie, The Mechanics of Engineering Soils, 5th Ed.,
6.6
H.Kennedy, External loads and foundations for pipes, J.Am Water Works
Tonbridge, Sept., 1969. Spon, London, 1969. Assn., March, 1971. LIST OF SYMBOLS B

trench width
Cc

load coefficient, embankment condition, positive projection case
Cd
load coefficient, trench condition
113
‘n
‘r Dord ES
E
load coefficient, embankment condition, negative projection case

load coefficient, rigid pavement

diameter

effective modulus of elasticity of soil

modulus of elasticity
h

pavement thickness or variable depth
H

depth of cover
He I

moment of inertia
k

modulus of subgrade reaction
K

ratio of lateral to vertical soil pressure
L M

bending moment
N

bending or deflection coefficient (subscripts t, b, s , x and y
depth of fill below plane of equal settlement
length of loaded area
refer to top, bottom and side moments and horizontal and vertical distortions respectively) pressure point load line load per unit length radius radius of stiffness settlement ratio wall thickness projection ratio vertical pressure W

lateral soil pressure
W

permissible ring load

permissible vertical load
w x

vertical load per unit length
Y

width of loaded area
W V
horizontal distance
6

angle of bottom support
A

deflection
Y

unit weight of soil
0

angle of friction between backfill and sides of trench

angle of friction of backfill material
0 V

Poisson’s ratio
1 1 4
CHAPTER 7
CONCRETE
PIPES
THE EFFECT OF BEDDING Nonpressure sewer or , ain pipes are loads, not internal pressures.
!signed to wit
m d external
Various standards specify the design load
per unit run of pipe for different classes and the pipes are reinforced accordingly.
The main stresses due to vertical external loads are the com
pressive stress at the haunches and the bending stresses at the crown, the bottom and the haunches.
The main stresses are caused by live loads, vertical
and horizontal soil loads, self weight and weight of water (internal water pressure and transient pressures are neglected for nonpressure pipes). Concrete pipes and the other rigid nonpressure pipes are normally designed to withstand a vertical line load while supported on a flat rigid bed.
The load per unit length required to fracture the pipe loaded thus is
called the laboratory strength.
Although the laboratory strength could be
calculated theoretically, a number of practical factors influence the theoretical load and experimental determination of the load is more reliable. (The tensile strength of concrete is very uncertain and the effect of lateral constraint of the supports may be appreciable). Alternative standard testing arrangements for pipes are illustrated in Fig. 7.1
. The strength is defined in British Standard 556 as that load which
will produce a crack 1/100 inch (0.25 mm) for unreinforced concrete pipes or 80% of the ultimate load for reinforced pipes, or 90% of the 1/1000 inch
(0.025 mm) wide crack load for prestressed cylinder type concrete pipes.
I
Ornm
15Ox 25rnrn rubber
2 EDGE
FIG. 7.1
3EDGE
3EDGE F O R ANY DIA UP TO l8OOmm
Standard crushing test bearings for rigid pipes.
115
The strength using the 3edge bearing test is usually 5 to 10% in excess of that for the 2edge bearing test.
Recently the 3edge bearing test
was standardized and this is now defined as the laboratory strength. Pipes laid in trenches are usually supported over a relatively wide length of arc by the bedding.
The strength is consequently higher than the
laboratory strength as the bending moments are less.
Table 6.3 summarizes
the theoretical bending moment coefficients for pipes supported over various angles
(defined in Fig. 8.4).
The ratio of field strength to laboratory strength is defined as the bedding factor.
Various types of bedding and the corresponding bedding factors
are listed below:TABLE 7.1
BEDDING FACTORS ~
Class A
120'
R.C. concrete cradle or arch
3.4
Class A
120°plain concrete cradle or arch
2.6 1.9
Class B
Granular bedding
Class C
Hand shaped trench bottom
1.5
Class D
Hand trimmed flat bottom trench
1.1
For rigid concrete pipes the lateral support of the side fill in the trench does not add noticeably to the strength. A factor of safety K of 1.25 to 1.5 is normally used with unreinforced concrete pipe.
When pipe is pressurized, it should be designed so that:
external load field strength
+
internal pressure bursting pressure
fcoL, which is usually fctL
=p
+(fCmL

fCOL)?
the case, then

+
2
and if
then
COL
2
(7.20a)
fcoL > fcmL
fctL
=
p*
+(fCOL
2
2
fCmL)

fcoL
2
fCmL (7.20b)
122
SHEAR STRESS
COMPRESSIVE STRESS
FIG. 7 . 3
Mohr circle for stress in core during winding.
It will be seen from Equs. 7.20 that the larger the longitudinal prestress, which is proportional to fcoL, the smaller is the principle tensile stress in the core f ctL' Longitudinal Stresses After Losses The longitudinal bars act to resist longitudinal bending of the pipe in the field. bedding.
The pipe acts as a beam spanning between uneven points in the
At this stage a certain amount of shrinkage and creep will have
occurred in the concrete and some longitudinal compressive stress will have been transferred to the coating. A s the longitudunal stress is not high, the concrete creep can u s u a l l y be neglected.
Shrinkage of the core does reduce
the tensile stress in the bars and the compressive stress in the core becomes
(7.21)
Once the pipe is in service, longitudinal bending stresses are added to the stress due to prestressing. fc3L
_ MD

21
The extreme fibre bending stresses are (7.22)
where M is the bending moment, and the effective moment of inertia of the section is
123
I
=
T (D 4  d 4) c4
+ (n,  1) As
(d + 2t
i
dS)*
8
(7.23)
Properties of Steel and Concrete The steel used for winding prestressed concrete pipe should have as high a yield stress as possible and a yield stress of 1 650 N/mmz(240 000 psi) is not uncommon.
This will ensure that after creep and shrinkage have taken
place, the remaining compressive stress in the concrete is fairly high.
The
steel stress will drop after winding, and it is normal to confine working stresses to less than 50% of yield stress. Steels with high yield stresses are often brittle and difficult to work with and in fact the ultimate strength may not be much higher than the yield stress. Care is therefore necessary in selecting the prestressing wire. The 6 modulus of elasticity of steel is approximately 200 000 N/mz(30 x 10 psi). The steel may creep slightly after prestressing but it is difficult to distinguish between concrete and steel creep between winding and works test, so
they are usually considered together. This loss in stress in the steel is
typically around 5% and occurs within a few hours after winding. Concrete cores are cast under vibration or centrifugally and 28 day cube crushing strengths of 60 N/mm2 (8 700 psi) are frequently achieved. High early strength (e.g. 50 N / m 2 or 7 200 psi)is as early as possible.
desirable for winding
Compressive stresses up to 50% of the cube strength
at the time are permitted during winding (any cracks will most probably heal).
Working compressive stresses should be confined to less than 1/3 of
the 28 day cube strength. The bending tensile stress in the core during winding should be less than about 10% of the cube strength at that age, and the calculated bending tensile stress in the core due to longitudinal bending in the field should be limited to about 5% of the cube strength in order to allow for unknowns and to prevent any possibility of cracks developing. No circumferential tensile stress is permitted in the core in the field for normal operating conditions but tension is sometimes permitted under transient conditions. The tensile stress in the coating should be less than about 10% of the cube strength, (This is for buried pipe. Exposed pipes may develop cracks and the tensile stress should be less than this.)
BS 4625 does not permit tensile stress in the core for normal operating plus backfill pressures, but permits a tensile stress of 0.747b if water
124
hammer pressure is included, and a tensile stress of 0.623
fi during works
hydrostatic test. (Where F is the crushing strength of 150 mm cubes at 28 days all in N / m 2 .) The modulus of elasticity of concrete increases with the strength and varies from 20 000 N/mm2 ( 3 000 000 psi) to 40 000 N/mm2 (6 000 000 psi). The creep coefficient, w, of concrete is defined by the equation creep strain where f
=
AL/L
w fc/Ec
=
is the average compressive stress during the time that creep occurs
and Ec is the final modulus of elasticity. w varies with time and the method of curing.
Creep is high for 'green' concrete and the rate of creepreduces
with time. w is approximately 0.3 two days after casting (at the time of factory testing the core) and is 1.3 three months after casting. Hence for the coating, the coefficient 1.3 is used to calculate creep, but for the core, the creep coefficient for the time between factory test and field conditions is 1.3  0 . 3
1.0.
=
It may be 30% higher for pipes not properly cured
or exposed to the elements in the field. Shrinkage of concrete also depends largely on the method of curing, 4 ty?ical values of shrinkage per unit length, v, varying from 10 to Example Calculate circumferential concrete and steel stresses during winding, after curing and under field conditions for the prestressed concrete pipe described below: Effective pipe length L tc
75 mm, coating tb
=
c/c.
S
=
4 m, bore di
=
0.00131 m2/m of pipe.
=
diameter , As
=
=
2 000 mm, core thickness
25 mm, steel winding 5 mm diameter at 15 mm Longitudinal wires 2 4 No. x 8 mm
0.00121 m2. Maximum internal pressure 0.8 N / ~ z . Vertical
loading due to soil and live load 0.04 N / m 2 , lateral soil pressure 0.5 x vertical
0.02 N / m m 2 .
=
Bottom support effectively over 30'
of arc. Neg
lect selfweight and weight of water inside. Steel: Longitudinal prestress N/mm2
Concrete:

Ecl
Es =
=
=
400 N/mm2. Winding prestress
200 000 N/mm2.
30 000 N / m m 2 ,
Ec2
Creep coefficient u =
I
n2
=
200 000/30 000
=
6.7
=
200 000/38 000
=
5.3
=
1 000
0.95.
38 000 N / m m 2 , Creep w
from factory test to field test. Wb n
=
=
=
1.3, shrinkaie v
1.0 =
125
Stresses due to winding:
1 000 x 0.0131 fco
=
f
=
so
=
16.5 N / m 2
0.075 + 0.00131 x 6.712 0.075
16.5
(7.2) =
880 N/mm2 (7.3)
0.00131
Works test of core: fcl
=
0.95 x 16.5
=
15.7 N / m 2
fsl
=
0.95 x 880
=
840 N / m 2
(7.4)
After curing:
y s 2
=
2+1 .o 104x 38 x 103(0.075 + 0.0252t1.3 ) + 0.075 x 15.7 x 1.0 2+1 .o 10 (1 + +) + (0.075 + 0.025 2+1.3)f5*3
0.00131
fs2
fb2
fc2
=
76 N / m 2
=
84076
= (
76 200 000

(7.9)
764 N / m 2
=
4 2 x 38 000
10
)
=
6.4 N / m 2 (7.10)
2+1.3
764 x 0.00131  6.4 x 0.025
11.2 N/m2
=
(7.11)
0.075
Under field loading: Transformed section
A
=
0.075 + 0.025 + (5.3  1) 0.00131 =
1
0.1 + 0.00131 (5.3  1) 0.0515
=
Centroid to outer surface Moment of inertia
I
e
(7.12)
0.12/2 + (5.3 1)(0.075 + 0.005/2)(0.00131)
=
Centroid to inner surface e
0.105 m2/m
0.1  0.0515
=
=
, ‘47.13)
=
0.0485
(7.14)
0*04853 + (0.075 + 0.005/2 0.0515)2
+
3 x (5.3 Stresses at base B : (tension


1)(0.00131)
=
6 4 85.9 x 10 m /m
ve for concrete stresses)
Tension due to net internal pressure:
(7.15)
126
ft
=

0.8 x 2  0.02 x 2.2
7.4 Nlmz
=
2 x 0.105 Bending moment coefficients from Table 6.3: For vertical load, Nb Net bending moment
=
0.235, and for horizontal load, Ns
=
0.235 x 0.04 x 2.212
=
0.00755
Stress on outer face stress on inner face
+
fb3
=
fb2

f
fc3
=
11.2

7.4
t



=
0.125
0.125 x 0.02 x 2.212
Meo I 0.00755 x 0.0515
=
Nlm2
85.9 x Steel stress:
e
=
0.075 + 0.005/2  0.0515
=
0.026 (On outer side of centroid)
Me fs3
=
f s 2 + n2(ft =
y)
764 + 5.3 (7.4

0.00755 x 0.026) 85.9 x
=
lou6
791
Nlm2.
Although the stresses at the base are usually the most severe, the stresses at other points on the circumference should be checked similarly, and checks should be done both with and without transient winding, testing and in the field. REFERENCES 7.1
Concrete Pipe Assn., Loads on Buried Concrete Pipes, Tech.Bulletin No.2, Tonbridge, Sept., 1962
7.2
Concrete Pipe A s s n . , Bedding and Jointing of Flexibly Jointed Concrete Pipes, Tech.Bulletin No.1, Tonbridge, Oct., 1967.
7.3
S.P. Timoshenko and S.WoinowskyKrieger, Theory of Plates and Shells,
7.4
Am.Concrete Pipe Assn., Design Manual
7.5
P.M.Ferguson, Reinforced Concrete Fundamentals, Wiley, N.Y., 1958.
7.6
BSCP 2007 Part 2, Reinforced and Prestressed Concrete Structures,
7.7
H.F.Kennison, Design of prestressed concrete cylinder pipe, J.Am.
2nd Edn., McGraw Kill, N.Y., 1959.

Concrete Pipe, Arlington,
1970.
BSI, London
.
1970.
Water Works Assn., 42 (Nov., 1950).
127
L I S T OF SYMBOLS
A

cross sectional area
d

inside diameter
D

outside diameter
e

distance from centre of gravity
E
modulus of elasticity
F I

K

factor of safety
L

length
M

bending moment
n

elastic modular ratio Es/Ec
q

shear stress
s

steel area per unit length of pipe
t

thickness creep coefficient for steel and concrete before factory test
V

W

creep coefficient
f
U
stress (compressive o r tensile) crushing strength of 150 m cubes at 28 days moment of inertia
shrinkage coefficient
Subscripts b

coating
C

core bending
t

L

longitudinal after winding
1

2

at time of laying
3

under field pressure
S
m 9
i 0
steel shear tensile initial at time of factory test
128
CHAPTER 8
STEEL AND FLEXIBLE PIPE INTERNAL PRESSURES The highest pressures a pipe has to resist are normally those due to internal fluid pressure.
The pressure is uniform around the pipe and there
are no bending stresses.
The general equations for the resulting stresses
in a hollow cylinder are:p.d.' Circumferential wall stress
Radial Stress Fr
=
1 1
Fw
=
 Podo2 do2  d.2
p.d.' 1 1
do2
+


Podo2 dS2
d., ' do2 (po d
2
diZdo2 (popi) 1 do2  d.' d2

pi)
 d.2
1 d2
(8.1)
(8.2)
where p is pressure, d is diameter at which the stress is sought, d. is internal diameter and d
is external diameter. The equations are for plain
stress, i.e. a cylinder free to expand longitudinally. For the particular case of no external pressure the circumferential stress is a maximum on the inner surface and is
F max W
=
d.2 + do2 d. + do
P 2t.

(8.3)
In practice the wall thickness is normally small in comparison with the diameter, and the wall thickness of a steel pipe designed to resist internal pressure is obtained from the formula t = pd 2f GJ
(8.4)
where t is the wall thickness p is the internal pressure d is the external diameter f is the yield stress of the steel G
is the design factor
J is the joint factor.
The external diameter is used i n preference to the internal diameter, as it partly corrects the inaccuracy in not using the precise Equ. 8.3.
123
The design factor G allows a safety margin.
What factor to use will
depend on the accuracy with which loads and transient pressure have been assessed, the working pressure, economics and the consequences of a burst. If the pipeline is protected against water hammer overpressure, a factor of
0.6 or even 0.7 is reasonable, but if there are many unknown loads, use of a factor of the order of 0.3 to 0.5 would be wiser. The joint factor J allows for imperfections in welded seams, and varies from 0.85 for furnace buttwelded joints to 1.0 for seamless pipes. Smallbore, high pressure pipes are designed to resist internal pressures only, but the larger the diameter and the the thinner the walls, the more important become the external loads. Vaporization of the fluid in the pipe may also be possible, in which case this internal pressure should be considered acting in conjunction with external loads. If the pipe is restrained longitudinally, a longitudinal tension of magnitude vF is induced where v is Poisson's ratio and F is the circumferential wall tension. TENSION RINGS TO RESIST INTERNAL PRESSURES Pipes which have to resist very high internal pressures may be strengthened by binding with hoops and spirals.
It is often difficult to
form pipes in the thick plates which would be required to resist some high pressures.
On the other hand it may be possible to form the pipe of thinner
plate than would be required for an unstrengthened pipe, and wind it with straps or rods. The rings required to resist internal pressures do not perform the same function as the stiffening rings required to resist collapse against external loads.
The winding to resist internal pressures acts in tension
and does not need to be as prominent as the rings to resist external load, which should increase the moment o f inertia of the longitudinal section through the pipe and therefore be as high as possible.
In fact the tension
rings, as they will be called, should be as flat and broad as possible to keep the distance between them to a minimum.
There will be high longitudinal
bending stresses and circumferential stresses in the pipe wall between rings if the rings are far apart.
In comparison, stiffening rings to resist ex
ternal loads are usually spaced a number of diameters apart.
Tension rings
may also be used to strengthen old pipes which are to be subjected to higher pressures than originally designed for.
130
Using the stressstrain relationship and the differential equations of equilibrium for cylinder shells under the action of radial pressure, Timoshenko derived an equation for the radial displacement of a cylinder subjected to a radial pressure and with tension rings at equispacing (Ref.8.1) (8.5) and
a
bs
=
(8.6)
where v is the Poisson's ratio (0.3 for steel), d is the pipe diameter, t is the wall thickness and s the centretucentre ring spacing. Let
XI
=
cosh a + cos a sinh a + sin a
(8.7)
x2
=
sinh a  sin a sinh a + sin a
(8.8)
cosh a  cos a Tinh a + sin a
(8.9)
sin (a/2) cosh (a/2) + cos (a/2) sinh (a/2) sinh a + sin a
(8.10)
3'
=
x4
=
Then the ring stress, Fr, is given by
(8.11) where A
=
cross sectional area of the ring.
The circumferential pipe wall stress, Fw, midway between rings is given by (8.12) and the longitudinal bending stress in the pipe under the rings, F is given b by
The stresses are indicated by Fig. 8.1, for v that as s is decreased the ring stress F have been anticiapated.
tends to
Note that for small A, F
ential wall stress of a plain pipe, pd/2t. ential pipe wall stress F
W
tends to
1
=
4
It may be shown
0.3.
pds/(ts + A) which could
tendst0 equal the circumfer
Also, for small
s,
the circumfer
pds/(ts + A), which would be expected, and
the longitudinal bending stress Fb tends to zero. For large ring spacing ( > approx. 2 &?I),
N
Circumferential wall stress m i d w a y between r i n g s
FIG. 8.1
Stresses associated with tension rings.
b.Ring s t r e s s
c . Bending stress under r i n g .
131
a.
132
F
1
tends to
1 + 0.91A/t,&
1.65
F tends to b and F
tends to
W
2t
t &/A
+ 0.91
Pd 2t
(8.14)
Pd 2t
(8.15)
i.e. that for a pipe without rings.
It may be observed by comparing F ,F and F
from Figs. 8.la, 8.lb w r b and 8.lc that the maximum stress for most practical ring sizes is in fact Fw, the circumferential pipe wall stress.
s/
is less than approximately 2.0.
Also, F
W
is only reduced if
In other words, the ring spacing
should be less than 2 m a n d the ring cross sectional area A should be of the same order of magnitude as the pipe wall longitudinal cross sectional area between rings, ts, to enable the rings to be of use. DEFORMATION OF CIRCULAR PIPES UNDER EXTERNAL LOAD For large diameter and flexible pipes under low internal pressures, the external load is frequently the critical one. Pipes may fail under external load by buckling, overstressing due to arching or bending~ cr excessive deflection. For elastic rings under plane stress subjected to vertical loads only, Spangler (Ref. 8 . 2 ) evaluated the bending moments and deflection at critical points around the circumference. The worst bending moments occur at the crown, the invert or the sides.
The bending moments per unit length are
given by an equation of the form
M
=
(8.16)
NWR
where R is the pipe radius, W is the vertical load per unit length and N is a coefficient. (Nt,Nb and Ns for top, bottom and sides respectively). The vertical and horizontal changes i n diameter are practically equal and opposite and are of the form
n
=
N WR~IEI Y is a coefficient.
(8.17)
where N Y The moment of inertia I per unit length of plain pipe wall is
I
=
(8.18)
t3/12
Figs. 8.2 and 8.3 give values of N
N
t’ b
and N
for the bending moments at
the top, bottom and side of the pipe respectively, and N for the vertical Y The coefficients are for a line load or a
deflections of the pipe diameter.
load distribution across the width of the pipe (W per unit length), and for different angles of bottom support B, as indicated in Fig. 8.4.
133
Angle of bottom support p degrees
FIG. 8.2 Wall bending coefficients for loaded pipe.
Angle of bottom support p degrees
FIG. 8.3 Deflection coefficients for loaded pipe.
134
Collapse of a steel pipe will probably not occur until the diameter has been distorted some 10 or 20 percent.
In practice deflections up to 5 percent
of the diameter are sometimes tolerated.
The deflection should normally be
limited to about 2 percent to prevent damage to linings, and for pipes with mechanical joints. The hydraulic properties, i.e. the crosssectional area and wetted perimeter are not affected noticeably for normal distortions.
L O A O W / U N I T L E N G T H OF PIPE
I
UPTHRUST
FIG. 8.4 Pipe loading and deflections. Effect of Lateral Support The lateral support of sidefill in a trench increases the strength of flexible pipes considerably and reduces deformations. Without lateral support to a pipe the ring bending stresses at the soffit and haunches or deflections would limit the vertical external load the pipe could carry.
But a
pipe in a compacted fill will deflect outwards laterally as it is loaded vertically thereby increasing the pressure of the sidefill against the sides of the pipe.
An equilibrium condition may be established with the vertical
load being transferred to the haunches by arch action as well as by ring action.
The stress due to the arch action is compressive s o that the load
which the pipe can carry is considerably higher than if the pipe were acting in bending.
In the extreme case, the lateral stress will equal the
vertical load stress and the pipe wall will be in pure compression, with
the stress equal to wd/2t.
w
where
=
(8.19)
Wld
If the pipe underwent no noticeable lateral distortion the load it could support would be determined by the bending strength plus whatever arch strength is given to the pipe by the soil pressure on the sides of the pipe i.e. the permissible vertical load per unit area on the pipe is w
=
+
Wb
w
(8.20)
where wb is the permissible bending load (limited either by ring bending stress or more likely by deflection). w
is the lateral soil pressure on
the sides of the pipe, which, conservatively, may be taken as the active soil pressure, but is usually greater than as indicated in subsequent equtions.
For sand, the active lateral pressure is approximately one third
of the vertical pressure due to soil dead load only and for clay it is approximately
3
the vertical pressure.
If the vertical load is greater than the sum of the ring load plus active lateral soil pressure, the pipe wall will deflect out laterally and increase the lateral pressure.
The horizontal stress will decrease away
from the pipe and Barnard, using elastic theory (Ref. 8.3)
suggests assuming
a triangular stress distribution with the horizontal pressure equal to total vertical pressure minus ring load at the pipe wall decreasing linearly to zero at 2.5d away from the pipe wall.
The corresponding lateral
deflection of each side of the pipe is AX/2
where E
=
1.25 (w  wb) d/E
(8.21)
is the effective modulus of elasticity of the soil.
The factor
1.25 should be increased as the lateral deflection increases, since the radial pressure increases as the radius of curvature decreases.
The factor
becomes 1.4 for a deflection of 2 percent of the diameter and 1.7 for a deflection of 5 percent. additional 25
The deflection also increases with time, and an
to 50 percent of the initial deflection can be expected
eventually. The relationship between stress and strain for the soil should be determined from laboratory triaxial consolidation tests.
The effective
modulus of elasticity of soil varies widely depending on soil type, degree of compaction or natural density, confining pressure, duration of loading and moisture content.
For example it may be as low as 2 N / m z for loose clay
or as high as 20 N / m 2 for dense sands. for loosely compacted fill, 5 N / m Z
The modulus is approximately 3 N / m 2
for fill compacted to 90% Proctor density
and 7 N / m 2 for fill to 95% Proctor density. Values higher than 100 have been
136
recorded for moist compacted sands. The ring load on the pipe is proportional to the elastic deflection as Taking N equal to 0.108, which corresponds to bottom Y and the load over the entire pipe width and putting
indicated by Equ. 8.17. support over 30' AY
AX
=
A
...
=
A , then solving for A/d from 8.17 and 8.21,
0.108wd3 8EI + 0.043Esd3

d
For plain pipe I
(8.22) t3/12, so one has an equation for deflection as a
=
function of diameter, loading. soil modulus and the ratio wall thickness/ diameter A d
.

0.108 0.67 E (t/d)3 + 0.043Es
(8.23)
The relationship between deflection and wall thickness for plain pipe is plotted in Fig. 8.5.
It will be observed that a steel pipe wall thickness
as low as J X of the diameter will be sufficient to restrain distortion to 2% provided the soil modulus is greater than 5 mPa. Pipes are sometimes strutted internally during backfilling of the trench to increase the vertical diameter and reduce the horizontal diameter. The lateral support increases when the struts are removed and the pipe tends to return to the round shape.
The vertical deflection and tendency to buckle
are consequently reduced considerably
STRESS DUE TO CIRCUMFERENTIAL BENDING
It is possible to compute wall stresses due to bending and arching. (Ref. 8.6).
If it can be assumed that the load is spread over the full width
of the pipe ( a
=
180°) and the bottom support is over 60°(
B
=
6 0 ' )
for
flexible pipe, then from (8.17) and Fig. 8 , 3 , AY
=
4 0.103wbd /8EI
Now from (8.21),
AX
=
(8.17b)
2.5(w  w )d/E b s
(8.21b)
Equating AY and AX, and solving for wb the ring load
wb
w1d3 = I/d3 + 0.006Es/E (8.24)
The bending moment in the wall, M, is due to ring load and is a maximum at the base and from (8.16) M
=
0.19~d2/2 hence bending stress f b b
=
Mr/I
(8.25)
137
where r i s the distance from the centre of gravity of the section to the extreme fibre (t/2 for plain wall pipe).
The balance of the load is taken
i n arch action of the pipe according to (8.20) so
0.006wEs/E
w a
= I/d3 + 0.006Es/E
(8.26)
The bottom wall hoop stress is fa
=
wad/2a
(8.27)
where a is the cross sectional area of wall per unit length (t for plain wall pipe). The total lateral compressive stress in the wall at the base is f
=
fb + fa, therefore permissible loading per unit area in terms of per
missible stress f is (I/d3 + 0.006Es/E)f w
= 0.1 r/d + 0.003 dEs/Ea
(8.28)
Thus the permissible vertical load w is a function of the permissible stress, the relative thickness t/d and the ratio of moduli of soil Es steel E . E
210 000 N/mm*, and f
=
to
The relationship is plotted in Fig. 8.5 for plain wall pipe, with =
210 N/mm2.
For thick pipes, the permissible load increases with wall thickness as more load can be taken in ring bending.
For thinner pipes, the deflection
becomes large so that the soil sidethrust increases until the pipe is in pure compression, and the limit is due to the wall hoop stress. Actually side wall hoop stress exceeds that at the base, so (8.27) plus (8.25) is not the limiting stress for high arching. On the same chart is plotted buckling load w
=
J'32EsEI/d3
(8.29a)
The buckling equation, proposed by C I R I A (Ref. 8 . 4 ) allows for lateral support. This may be compared with an alternative equation investigated by the Transport and Road Research Laboratory (which is found to overpredict wc): w
=
(16EsZEI/d3)1 /3
(8.29b)
The load wd giving a deflection of 2% in the diameter is also plotted on the chart from (8.23).
The chart thus yields the limiting criterion for any
particular wall thickness ratio.
The lowest permissible load w is selected in
each case from the chart by comparing deflection, overstressing and buckling lines for the relevant t/d and E
.
13 8
Similar charts should be plotted where stiffening rings are used and where alternative material moduli and permissible stresses apply. More General Deflection Equations _ I 
Spangler (Ref 8.2) allowed €or lateral support to the pipe due to the backfill in a more theoretical way than Barnard.
The equation he derived
for vertical deflection is: UZWd3 8 EI + 0.06Esd3
A =
(8.30)
This corresponds to (8.22) if UZ is substituted for N and a value of 0.15 Y is assumed for N in the denominator (instead of 0 . 1 0 8 in 8 . 2 2 ) Y Here U = soil consolidation time lag factor (varies from 1.0 to 1 . 5 ) .
Z
=
bedding constant (varies from 0.11 for point support to
I
=
moment of inertia of pipe wall per unit length.
0.083 for bedding the full width of pipe), normally taken as 0 . 1 . passive resistance modulus of sidefill.
Es =
The pressure inside a pipe may also contribute to its stiffness. Due to the fact that the vertical diameter is compressed to slightly less than the horizontal diameter, the vertical upthrust due to internal pressure becomes greater than the sidethrust by an amount of 2pA which tends to return the pipe to a circular shape. A more general expression for vertical deflection thus becomes
a =
UZWd3 8 E T + 0.05Esd3 + 2 UZpd3
om2 FIG. 8.5
0.01
0005 Relotiw thickness
tb
Permissible load on plain pipe.
(8.31)
0.02
0.03
139
STIFFENING RINGS TO RESIST BUCKLING WITH NO SIDE SUPPORT Morley (Ref. 8.5) developed a theory for the buckling of stiffened pipes under uniform external pressure.
The theory often indicates stiffening
ring spacings wider than is considered necessary in practice.
The theory
also neglects the possibility of failure under combined internal and external pressures and bending.
The equations do, however, yield an indication
of stiffening ring spacing. Using an analogy with a strut, Morley developed an equation which indicates the maximum vertical external pressure, w, which a cylindrical shell can take without buckling.
The equation allows for no axial expansion, and
assumes the wall thickness is small in comparison with the diameter. v is Poisson's ratio,
w = 24 1  v* For plain pipe, I
EI d3
(8.32)
t3/12, so
=
Experiments indicated a permissible stress 25% less than the theoretical so
for steel,
w
=
t
1.65 E ( J ) ~
(8.34)
For thickwalled tubes, the collapse pressure will be that which stresses the wall material to its elastic limit (w
=
2ft/d) whereas for intermediate
wall thickness, failure will be a combination of buckling and elastic yield. An empirical formula indicating the maximum permissible pressure on a pipe
of intermediate thickness is w = 2ft d
/ (1
fd2  5)
(8.35)
where f is the yield stress. The external load may be increased if stiffening rings are used to It was found by experiment that the collapse load, w, i s
resist buckling.
inversely proportional to the distance between stiffening rings, s, if s is less than a certain critical length, L. From experiment L
=
so if w
=
2E (t/d)3
w
=
$ 2E
(t/d)3
1.73
(8.36)
r$
for stiffening pipe, then for stiffened pipe, =
3.46
5
(8.37)
1 4 0
The actual permissible stress is less than the theoretical due to imperfections in the material and shape of pipe,
so
the practical ring spacing
is given by
(8.38)
If the full elastic strength of the pipe is to be developed to resist vertical external pressures i.e. w
sd
=
2ft , then the ring spacing should be d
= E E
(8.39)
f
t
Rings will only be of use if w > 1.65E (J )3. It should also be ascertained 2ft that w l , which is the elastic yield point. d If w
=
2ft/d, then rings will only be of use if
t/d
209
Break horse power (BHP) or shaft power required by the pump and pump efficiency E are often indicated on the same diagram (see Fig. 12.7). HQ
The duty
curve may alter with the drive speed and could also be changed on any pump
by fitting different impeller diameters. then.
The BHP and E curves would also change
The resultant head is proportional to N2 and D Z , whereas the discharge
is proportional to N and D. The required duty of a pump is most easily determined graphically. The pipeline characteristics are plotted on a head  discharge graph.
Thus Fig. 12.8
illustrates the head (static plus friction) required for different Q's, assuming (a) a high suction sump water level and a new, smooth pipe, and (b) a low suction sump level and a more pessimistic friction factor applicable t o an older pipe.
line.
A line somewhere between could be selected for the duty
The effect of paralleling two or more pumps can also be observed from
such a graph.
H
2 in parallel
high sump level and new pipe
0
FIG. 12.8 Pipeline and pump characteristics
2 1 0
Pumps is series are more difficult to control. booster pump station (Fig. 12.9)
Frequently, however, a
is required along a pipeline to increase
capacity or to enable the first section of pipeline to operate within a lower design pressure.
In such cases the pump curves are added one above the
other, whereas for pumps in parallel, the discharge at any head is added, i.e. the abscissa is multiplied by the number of pumps in operation. Booster pumps may operate inline, or with a break pressure reservoir or sump on the suction side of the booster.
In the latter case a certain
balancing volume is required in case of a trip at one station. The hydraulic grade line must also be drawn down to reservoir level.
In the former case
water hammer due t o a power failure is more difficult to predict and allow for.
Hydraulic qrade line
FIG. 12.9 Location of booster pump station. MOTORS The driving power of a pump could be a steam, turbine, water wheel or most commonly nowadays, an electric motor.
The rotational speed of the pump
is controlled by the frequency of alternating current. approximately N
=
The pump speed is
60Hz divided by the number of pairs of poles, where N is the
pump speed in revolutions per minute and Hz is the electrical frequency in cycles per second or Hertz. the rotor and starter.
There is a correction to be made for slip between
Slip can be up to 2 or 5 percent (for smaller motors).
21 1
Three types of AC motors are commonly used for driving pumps: (a)
squirrel cage induction motors (asynchronous)
(b)
wound rotor (slipring) induction motors (asynchronous)
(c)
synchronous induction motors. Squirrel cage motors are simple in design, robust, economic and require
minimum maintenance. types.
They are generally more efficient than the other two
Their disadvantage is that they require large starting currents.
However if this can be tolerated, a squirrel cage motor with directon line starter offers the most economical and reliable combination possible.
If
directon line starting is not permitted, it is usual to employ either stardelta or autotransformer starters. Alternative starters compare as follows:Type of Starter
Starting Current (proportion of full load)
Starting Torque (full load
= 1)
Direct  on
4.5 to 7.0
0.5 to 2.5
Star  Delta
1.2 to 2.0
0 . 3 to 0.8
Auto  transformer
as required
as required
Slipring induction motors are started by means of resistance starters and have the advantage that they can be smoothly brought up to full speed with relatively low starting current.
They are normally used when squirrel
cage motors are impermissible. Synchronous motors are used on large installations. They have the advantage of a high load factor.
They may be either induction motors or salient pole
motors, They run at constant synchronous speed.
Starting can be by directon
line, autotransformer, resistance or reactor starters depending mainly on the permitted starting current. Variable speed motors are more expensive and less efficient than constant speed motors.
Belt drives on smaller units are possible although commutator
motors, or motor resistances on slipring motors are used as well.
Thyristor
starters and variable speeds are now being used with savings in cost for small motors.
Pole changing can also be used for selecting two speeds for
squirrel cage motors.
PUMPSTATIONS The cost of the structures to house pumps may exceed the cost of the machinery.
While the design details cannot be covered here the pipeline eng
ineer will need to concern himself with the layout as he will have to install the suction and delivery pipework to each pump with control valves.
The
capacity of the suction sumps will effect the operation of the system, and there may be water hammer protection incorporated in the station. A more complete discussion of pumpstation layouts is given in Ref. 12.9.
21 2
The design of the sump and pump inlet pipework has an important bearing on the capacity of a pipeline.
erational efficiency.
Head losses into the pump can effect its op
Air drawn in by vortices or turbulence could reduce
the capacity of the pipeline considerably.
The awareness of the requirements
of the entire pumping system therefore should be the duty of the pipeline engineer. REFERENCES 12.1
A.J.Stepanoff, Centrifugal and Axial Flow Pumps, Theory, Design and Application, 2nd Ed., Wiley, N.Y.,
1957, 462pp.
12.2
Institution of Water Engineers, Manual of British Water Engineering
12.3
H.Addison, Centrifugal and Other Rotodynamic Pumps, 2nd Ed., Chapman and
Practice. Vol.11, Engineering Practice, 4th ed., 1969, London. Hall, London, 1955, 530p. 12.4
1.J.Karassik and R.Carter, Centrifugal Pumps, F.W.Dodge,N.Y., 1 9 6 0 , 4 8 8 ~ ~
12.5
A.Kovats, Design and Performance of Centrifugal and Axial Flow Pumps and Compressors, Pergamon Press, Oxford, 1964, 468pp.
12.6 KSB, Pump Handbook, Klein,Schanzlin 12.7
&
Becker, Frankenthal, 1968, 183pp.
N.B.Webber, Fluid Mechanics for Civil Engineers, Chapman and H a l l , London, 1971, 340pp.
12.8
E.Grist, Nett positive suction head requirements for avoidance of unacceptable cavitation erosion in centrifugal pumps, Proc.Cavitation conference, Instn.Mech.Engrs.,London,
1974, p153162.
12.9 A.C.Twort, R.C.Hoather and F.M.Law, Water Supply, Edward Arnold, London, 1974, 478pp. LIST OF SYMBOLS constants
A,B BHP
break horsepower
b
width
D
diameter
E
efficiency
g H
head(subscript a
m
mass per unit time
gravitational acceleration p


atmospheric, e  turbulence, f  friction
inlet and s  elevation).
2 1 3
N
pump rotational speed
NPSH
net positive suction head
NS
specific speed
P
pressure
P
power
Q
discharge rate
r
radius
T
torque
t
radial component of velocity
U
peripheral velocity
V
velocity
W
unit weight of water pg. (9800 Newtons per cubic metre)
P
unit mass of water
a, B
angles
2 14
GENERAL REFERENCES AND STANDARDS BRITISE STANDARDS S t e e l Pipes
BS
534
S t e e l P i p e s , F i t t i n g s and S p e c i a l s
778
S t e e l P i p e s and J o i n t s
1387
S t e e l t u b e s and t u b u l a r s f o r s c r e w i n g
1965Pt 1 B u t t  w e l d i n g p i p e f i t t i n g s . Carbon s t e e l 2633
Class 1 a r c welding of f e r r i t i c steel p i p e s
2910
R a d i o g r a p h i c e x . welded c i r c . b u t t j o i n t s i n s t e e l p i p e
3601
S t e e l p i p e s and t u b e s
3602

carbon s t e e l  o r d i n a r y d u t i e s
S t e e l p i p e s and t u b e s f o r p r e s s u r e p u r p o s e s . Carbon s t e e l : h i g h d u t 6 e s
3603
S t e e l p i p e s and t u b e s
3604
Low and medium a l l o y s t e e l p i p e s
3605
S t e e l p i p e s and t u b e s f o r p r e s s u r e p u r p o s e s .
f o r p r e s s u r e purposes
Carbon and a l l o y s t e e l p i p e s
Austenitic s t a i n l e s s steel
Cast I r o n P i p e 781
C I s p i g o t and s o c k e t p i p e s
782
C I s p i g o t and s o c k e t p i p e s

pipes fittings
143
M a l l e a b l e C I screwed p i p e s
416
C I s p i g o t and s o c k e t s o i l , w a s t e and v e n t p i p e s
437
C I s p i g o t and socket d r a i n p i p e s and f i t t i n g s
1130
C I d r a i n f i t t i n g s  s p i g o t and s o c k e t
121 1
C a s t (spun) i r o n p r e s s u r e p i p e s
1256
M a l l e a b l e C I screwed p i p e s
2035
C I f l a n g e d p i p e s and f i t t i n g s
4622
Grey i r o n p i p e s and f i t t i n g s
Wrought I r o n P i p e 788
Wrought i r o n t u b e s and t u b u l a r s
1740
Wrought s t e e l p i p e f i t t i n g s
Ductile Iron Pipe
4772
D u c t i l e i r o n p i p e s and f i t t i n g s
2 1 5
Asbestos Cement Pipe BS
486
A C pressure pipes
582
A
C soil, waste and ventilating pipes and fittings
2010Pt4 Design and construction of A C pressure pipes in land 3656
A C sewer pipes and fittings
Concrete 556
Concrete pipes and fittings
4101
Concrete unreinforced tubes and fittings
4625
Prestressed concrete pipes (inc. fittings)
Clay Pipe 65
Clay drain and sewer pipes

539
Clay drain and sewer pipes
540
Clay drain and sewer pipes
1143
Salt glazed ware pipes with chemically resistant properties
1196
Clayware field drain pipes
fittings
Plastic and Other Pipe 1972
Polythene pipe (type 3 2 ) for cold water services
1973
Spec.polythene pipe (type 4 2 5 ) for general purposes
3284
Polythene pipe (type 5 0 ) for cold water services
3505
UPVC pipes (type 1 4 2 0 ) for cold water supply
3506
UPVC pipe for industrial purposes
3796
Polythene pipe (type 5 0 )
3867
Dims. of pipes O.D.
4346
Joints and fittings for UPVC pressure pipe
4514
UPVC soil and ventilating pipe
4660
UPVC underground and drain pipe
4728
Resistance to constant internal pressure of thermoplastic pipe
2760 CP312
Pitch fibre pipes and couplings Plastic pipework
Insulation 1334
Thermal insulation
4508
Thermally insulated underground piping systems
CP3009
Thermally insulated underground piping systems
2 16
Valves Taps and v a l v e s f o r w a t e r
1010
BS
1212Pt2 B a l l v a l v e s

diaphragm t y p e
1218 & 5163(m) S l u i c e v a l v e s Valves f o r domestic p u r p o s e s
1415 1952
&
5154(m) Copper a l l o y g a t e v a l v e s 1953
Copper a l l o y check v a l v e s
2060
Copper a l l o y s t o p v a l v e s
2591
G l o s s a r y of v a l v e s
3464 5150(m) C I wedge and d o u b l e d i s c v a l v e s 3948
&
5151(m) C I p a r a l l e l s l i d e v a l v e s 3952
&
5155(m) C I b u t t e r f l y v a l v e s 3961
&
5152(m) C I s t o p and check v a l v e s 4090
&
5153(m) C I check v a l v e s 4133 & 5157(m) Flanged s t e e l p a r a l l e l s l i d e v a l v e s 4312
Stop and check v a l v e s
5156(m) Screwdown diaphragm v a l v e s 5 158 (m) P l u g v a l v e s 5159(m) B a l l v a l v e s Jointing
10
F l a n g e s and b o l t i n g f o r p i p e s e t c .
21
Threads
1737
J o i n t i n g materials and compounds
1821
Oxyacetylene w e l d i n g o f s t e e l p i p e l i n e s
1965
Butt welding
2494
Rubber j o i n t r i n g s
3063
Dimensions of g a s k e t s
4504
Flanges and b o l t i n g f o r p i p e s e t c .
Miscellaneous
1042
Flow measurement
1306
Nonferrous p i p e s f o r steam
217
BS
1553
Graphical symbols
1710
Identification of pipelines
2051
Tube and pipe fittings
2917
Graphical symbols
3889
Non destructive testing of pipes and tubes
3974
Pipe supports
4740
Control valve capacity
SOUTH AFRICAN BUTEAU OF STANDARDS Steel Pipe SABS 62
Steel pipes and fittings up to 150 mm
719
Electric welded low carbon steel pipes
720
Coated and lined mild steel pipes
Cast Iron Pipe 509
Malleable C I pipe fittings
746
C
815
Shouldered end pipes, fittings and joints
I soil, waste, water and vent pipes
Asbestos Cement Pipe 546
C I fittings for A C pressure pipes
721
A C soil, waste and vent pipes and fittings
946
A C pressure pipe  constant internal diameter type
286
A C pressure pipes
819
A C sewer pipes

constant outside diameter type
Concrete Pipe 676
R C pressure pipes
677
Concrete nonpressure pipes
975
Prestressed concrete pipes Structural design and installation of precast concrete
902
pipelines Glazed Earthenware Pipe 559
Glazed earthenware drain and sewer pipes and fittings
21 6
Plastic Pipe UPVC sewer and drain pipe fittings
SABS791 921
Pitch impregnated fibre pipes
966
UPVC pressure pipe
967
UPVC soil, waste and vent pipes
997
UPVC pressure pipes for irrigation
01 12
Installation of PE and UPVC pipes
533
Black polyethylene pipes
Valves 144
C I single door reflux valves
191
Cast steel gate valves
664
C I gate valves
AEERICAN WATER WORKS ASSOCIATION
Fabricating electrically welded steel water pipe
AWWA C201
Mill type steel water pipe
c202 C203
Coaltar enamel protective coatings for steel water pipe
C205
Cement mortar protective coatings for steel water pipe of sizes 30" and over
C206
Field welding of steel water pipe joints
C207
Steel pipe flanges
C206
Dimensions for steel water pipe fittings
C300
Reinforced concrete water pipe

Steel cylinder

Steel cylinder

Noncylinder type 
Type  not prestressed Reinforced concrete water pipe
C301
Type

prestressed
Reinforced concrete water pipe
C302
not prestressed C600 54T C602
Installation of C I watermains Cement mortar lining of water pipelines in place (16" and over)
APERICAN PETROLEUM INSTITUTE API
Std.5A Spec. for Casing, Tubing and Drill Pipe Std.5ACSpec. for Grade C

75 and C95 Casing and Tubing
Std.5AXSpec. for High Strength Casing and Tubing Std.5L Spec. for Line Pipe
2 1 9
API
Std.5LA
Spec. for Schedule 5 Alum.Alloy Line Pipe
Std.5LP
Spec. for Thermoplastic Line Pipe Spec. for Glass Fibre Reinforced Thermosetting Resin
Std.5LR
Line Pipe Spec. for Spiral Weld Line Pipe
Std.5LS Std .5LX
Spec. for High Test Line Pipe
RP5CI
Care and Use of Casing, Tubing and Drill Pipe
RP5LI
Railroad transport of Line Pipe Transmission
RP5L2
Internal Coating of Line Pipe for Gas
RP5L3
Conducting Drop Weight Tear Tests on Line Pipe
Bul.5C2
Performance Properties of Casing, Tubing and Drill Pipe
Bul.5TI
Nondestructive Testing Terminology
AMERICAN SOCIETY FOR TESTING MATERIALS Concrete Pipes ASTM C14
Concrete Sewer, Storm Drain and Culvert Pipe
C76
Reinforced Concrete Culvert, Storm Drain and Sewer Pipe
C118
Concrete Pipe for Irrigation or Drainage
C361
Reinforces Concrete LowHead Pressure Pipe
C412
Concrete Drain Tile
c443
Joints for Circular Concrete Sewer and Culvert Pipe with Rubber Gaskets
c444
Perforated Concrete Pipe
c497
Determining Physical Properties of Concrete Pipe or Tile
C505
Nonreinforced Concrete Irrigation Pipe and Rubber Gasket Joints
C506
Reinforced Concrete Arch Culvert, Storm Drain and Sewer Pipe
C507
Reinforced Concrete Elliptical Culvert, Storm Drain and Sewer Pipe
C655
Reinforced Concrete Dload Culvert, Storm Drain and Sewer Pipe
Steel Pipes ASA
B36.10 Steel pipes
220
Cast Iron Pipes A142
C I pipes
A377
C I pipes
A121.1 C I p i p e s A s b e s t o s Cement P i p e s
C296
A C Pressure Pipes
C500
A C Pressure Pipes
C428
A C P i p e s and F i t t i n g s f o r Sewerage and D r a i n a g e
P l a s t i c Pipes D2241
U n p l a s t i c i s e d PVC p i p e s
USDI BUREAU OF RECLAMATION S t a n d a r d Spec. f o r R e i n f o r c e d C o n c r e t e P r e s s u r e P i p e
221
BOOKS FOR FURTHER READING
1
H.Addison. A treatise on Applied Hydraulics, Chapman Hall, London, 1964
2
M.L.Albertson,J.R.Barton
3
American Concrete Pipe Association, Design NanualConcrete Pipe,
and D.B.Simons, Fluid Mechanics for Engineers,
PrenticeHall, 1960 Arlington, 1970 4
Am.Soc.Civi1 Engs. and Water Polln.Contro1 Federation, Design and
5
Am.Water Works Assn., Steel Pipe
6
H.S.Bel1, (Ed.), Petroleum Transportation Handbook, McGraw Hill,N.Y.,1963.
7
R.P.Benedict, Fundamentals of Pipe Flow, Wiley Interscience, 1977, 531pp.
Construction o f Sanitary and Storm Sewers, Manual 37,N.Y., 1970

Design and Installation, Manual Ml1,
N.Y., 1964
8
Bureau of Public Roads, Reinforced Concrete Pipe Culverts  Criteria for Structural Design and Installation, U.S.Govt.Printing Office, Washington, DC, 1963.
9
N.W.B.Clarke, Buried Pipelines

A Manual of Structural Design and
Installation, MacLaren and Sons, London, 1968.
10
Colorado State Univ.,Control of Flow in Closed Conduits, Proc., Inst., Fort Collins, 1971.
11
S.Crocker and R.C.King, Piping Handbook, 5th Ed., McGraw Hill, 1967.
12
C.V.Davis and K.E.Sorensen, Handbook of Applied Hydraulics, 3rd Ed., McGraw Bill, N.Y., 1969.
13
E.Holmes, Handbook of Industrial Pipework Engineering, McGraw Hill, N.Y., 1973.
14
1nstn.Water Engs., Manual of British Water Engg. Practice, 4th Ed., London, 1969.
15
C.T.Littleton, Industrial Piping, McGraw Hill, 1962, 349pp.
16
W.L.Martin, Handbook of Industrial Pipework, Pitman, London, 1961.
17
C.B.Nolte, Optimum Pipe Size Selection, Trans Tech Publications, 1978,297~~.
18
H.Rouse, Engineering Hydraulics, Wiley,N.Y., 1961.
19
V.L.Streeter (Ed), Handbook of Fluid Dynamics, McGraw Hill, N.Y., 1961.
20
A.C.Twort,R.C.Boather and F.M.Law, Water Supply, 2nd Ed., A.Arnold,
London, 1974. 21
J.H.Walton, Structural Design of Vitrified Clay Pipes, Clay Pipe Development Association, London, 1970.
22 2
APPENDIX SYMBOLS FOR P I P E FITTINGS GENERAL ELBOW
A. 1 1 E3
SLEEVED TEE FILLET WELDEDTEE
I
CROSS OVER
BELLMOUTH
D
I
BEND JKKETED FRONT VIEW OF TEE BACK VIEW OF TEE HANGER SIMPLE SUPPORT CHANGE IN OIA
TAPER JOINTS FLANGED ELECTRICALLY INSULATED FLEXIBLE
+
+I
ELECTRICALLY BONDED B U T WELD SCREWED
SWIVEL
__(
SOCKET
EXPANSION
€ 
SPIGOT
+
END CAP
S L E E V E COUPLING
e
4;1 + +
SOCKET
D
VALVES BUTTERFLY
ISOLATING WEDGE REFLLM
GLOBE
ROTARY PLUG
DIAPHRAGM
AIR
RELIEF
MISCEUANEOUS HYDRANT
STRAINER
FLOW INDICATOR
SPRAY VENT
SURFACE BOX DRAIN MOTOR
IA
NEEDLE
GATE
7 0
PLATE BLIND HANDWHE EL
w 6a
2 23
PROPERTIES OF P I P E SHAPES
AREA
= T r ( 0 2  d2)%=TIot
for small t
MOMENT OF INERTIA ABOUT DIAMETER =
6( D L 
d') ~ $ 0 ' ~ $03t
M O M E N T O F INERTIA ABOUT CENTRE $(O*d')
A R E AT* L a3b MOMENT O F I N E R T I A
ABOUT AXIS c d . n
MOMENT OF INERTIA ABOUT C E N T R E = ~ ~ P ~ ; (  = ~ + ~ ~ ~
PROPERTIES OF WATER
Temperature
Specific Mass
Kinematic Viscosity
Bulk Modulus
Vapour Pressure
sq ft/sec
N/mm2
psi
1.79x10+
1.93~10~
2 000
290 000
0.6~10~
0.09
1.31~10~
1.41~10~
2 070
300 000
1.2~10~
0.18
F0
kg/m3
l b / c u ft
m2 / s
0
32
1 000
62.4
10
50
1 000
62.4
C0
N/mm2
psi
20
68
999
62.3
1 .01x106
1.09~10~
2 200
318 000
2.3~10~
0.34
30
86
997
62.2
0.8 1 x1OV6
0.87~105
2 240
325 000
4.3~10~
0.62
PROPERTIES OF PIPE MATERIALS
Coef. of exp.per Clay
5x10
OC
Asbestos cement Cast Iron
8.5~10~
Mild Steel
pt.OC
Modulus of elasticity
psi
NIlTUTlz
Yield Stress
Tensile Strength
N/mz
psi
N/m2
psi
70
10 000
2.1
300
Poisson's ratio
6
10x1o6 6 8.5~10
Concrete
Softening
14 00040 000 24 000
6 6 2x10 6x10 6 3.5~10
17
2 500
0.2
100 000
15x1O6
150
22 000
225
23 000
0.25
11.9x106
210 000
3 1 ~ 1 0 ~
210
30 000
330
48 000
0.3
11.9x106 6 40x10
210 000
6 31x10
1 650
240 000 1 730
250 000
3
High Tensile Steel Pitch Fibre Polyethylene
160x106
60
138
20 000
120x106
120
700
0.1x10
400
14
2 000
5
700
24
3 500
8
1 100
17
2 500
12
1 800
52
7 500
Polyethylene (high density) PVC UPVC
50x1 06 6 50x10
80
3 500
6
6 0.5~10
0.38
226
CONVERSION FACTORS Length
1 inch
25.4 mm
=
1 ft = 0.3048 m 1 mile Area
1 sq inch
=
1 acre
2.47 ha
=
1 sq ft 1 ha Volume
1.61 km
=
644 mm2
0.0929 mz 4 10 m2
=
=
35.31 cu f t
1 m3
=
1 gal.(imperial)
1 US gal. 1 barrel =
4.54 litres
=
3.79 litres
=
42 US gal.
=
159 litres
Speed
1 ft/sec
Acceleration
32.2 ft/sec2
Discharge
1 mgd (imperial)
1 mph
0.3048
=
m/s
1.61 km/hr.
=
35.3 cusec
0.981
=
1 lb
Pressure
145 psi
(1 Newton
=
1 t/s
4.45 N
1 kg x 9.81 m/s2)
=
1 MPa
=
14.5 psi 778 ft lb.
=
1 MN/m2
1 bar
=
=
1 Btu
= 252 calories
1 calorie
=
4.18 Joules
550 ft lblsec = 1 HP
1 HP Kinematic Viscosity
=
1 slug (US)
=
1 lb. force
Power
=
0.746 kW
1 sq ft/sec =
=
929 stokes
0.0929 m2/s
Absolute or dynamic viscosity
1 centipoise = 0.001 kg/ms
Temperature
Fo
Absolute temperature Ro(Rankine)
=
FO(Fahrenheit)
+ 460
Absolute temperature Ko(Kelvin)
=
Co(Centigrade)
+ 273
71
=
3.14159
e
=
2.71828
=
g
1 m3/s
=
Force
1 Btu
=
0.454 kg
=
32.2 lb.
Energy
m/s2
1.86 cusec
=
13.2 gpm (imperial) Mass
= 35 imp.ga1.
32 + 1.8Co
=
1 N/mm2
227
AUTHOR
INDEX
Abrarnov,N. 14
Glass,W.L. 100
Addison,H. 212, 221
Goodier, J.N. 160
Albertson,M.L. 25, 221
Grist, E. 205, 272
52
Appleyard, J . R . 100
Avery, S . J .
HardyCross, 28, 47, 49, 51 Hassan, D.R. 160
Barnard, R.E. 112, 135, 142
HazenWilliams, 17, 22
Barton,J.R. 25, 221
Hoather,R.C. 212, 221
Bell, H.S. 221
Holmes,E. 221
Bernoulli, 15, 16, 175 Berthouex, P.M. 10, 13 Boucher,P.L. 180 Boussinesque, 109 Buras, N. 52
Cassie, W.E. 110, 180 Cates, W.H. 155, 199 Chapton, H.J. 160 Chezy, 17
James,L.D. 13 Joukowsky, 55 Kalinske, A.A. 92, 95, 100 Kally, E. 50, 52 Karrassik,I.J. 205, 212 Kennedy, H. 110
Clarke, N.W.B. 101, 112, 221 Crocker, S.
Jacobsen, S. 143 Johnson, S.P. 86
Capper, P.L. 112, 180
Cole, E . S .
Isaacs, L.T. 32, 51
199
Kennedy,J.F. 67, 86 Kennison, H.F. 126
160, 221
Cross,H. 5 1
King, C.L. 160 King, R.C. 160, 221
Dantzig, G.B. 52
Kinno,H. 67, 82, 86
Darcy, 17, 22, 27
Kovats, A. 205, 212
Davis, C.V. 221
Lai, C. 86
Deb, A.K. 51, 52 Denny,D.F. 90, 100 Diskin, M.H. 23, 25
Lam, C.F. 52 Law, F.M. 212, 221 Lescovich, J.E. 180 Littleton, C.J. 221
Ervine,D.A. 100
Ludwig, H. 61, 85 Fok, A.T.K. 82
Lupton, H.R. 85
Fox, J.A. 100 Manning, 17, 23
Froude, 95 Ferguson, P.M.
126
Mark,R. 112 Marks,P. 100 Marston, 101
228
Martin, W.L. 221
Stepanoff,A.J. 212
Mills, L.G. 3 2 , 51
Stephenson,D. 5 2 , 8 6 , 1 0 0 , 1 4 2 , 1 6 0
Mohr, 121
Streeter,V.L. 8 6 , 2 2 1
Moody, 22
Suss,A. 160
Morley, A. 142
Swanson, H.S. 147, 160
Morrison, E.B. 180
Sweeten,A.E. 180
Nelson, E.D. 160
Timoshenko, S.P. 1 2 6 , 1 4 2 , 160
Newmark, 110
Twort, A . C . 212, 221
Newton, 15, 53 Nikuradse, 1 9 , 2 0 Nolte,C.B. 221 Parmakian,J. 6 7 , 7 1 , 7 4 , 8 2 , 1 0 0 , 1 8 0 Pearson, F.H. 161 Paul, L. 1 7 9
Van der Veen, B. 5 2 Walton, J.H. 221 Watson, M.D. 2 5 Webber, N.B. 212
Poisson, 1 1 1 , 1 2 9 Proctor, 135, 183 Prosser,M.J. 9 0 , 100 Reitz, H . N .
Uhlig, H.H. 199
199
Weisbach, 22 White, 2 0 , 22 White, J . E .
13
Wilkinson, W.J. 160
Reynolds, 1 7 , 2 3
Wisner, P. 9 2 , 100
Reynolds, G.M. 199
WoinowskiKrieger, S. 1 2 6 , 1 4 3
Rich, G.R. 8 6
Wylie, E.B. 8 6
Riddick, T.M. 1 9 9 Roark,R.J. 160 Rouse, H. 160 Scharer,H. 161 Schlichting,H. 2 5 Schneider, W.R. 1 9 9 Schwartz, H.I. 1 9 9 Schweig,Z. 5 2 Simons, D.B. 25, 221 Sorenson, K.E. 221 Sowers, G.F. 1 9 9 Spangler, M.G. 102107,112,142
228
SUBJECT
INDEX
Absorption
88
Casing 202
Active soil pressure, 104, 135,172
Cast iron 162
Actuator 165
Cathodic protection, 192 Cavitation 77, 88
Adiabatic 7 8 Aesthetics 2 Air pocket, 88, 95,
Celerity 5 9 170
valves 77,96,169 vessel 77 Anode, sacrificial 193
Centrifugal f l o w 202 force 162 Characteristics method 6 0 pump60, 66, 202
Appurtenances 162, 2 0 1
Chemicals 1
Asbestos cement 163
CIRIA 137
Axial expansion 159
Clamp 187 Clay 174
Backfill 101,183 Barge 186 Beam 157 Bearing test 115 Bedding 108, 183 Bedding factor 134, 184 Bend l o s s 2 4 meter 177 stress 156, 171 Bending 132,136,139,157 Binary number 178 Bitumen 191 Boning 182 Boundary layer 18 Bracing t 4 5 Branch 59, 145 Bridge 185 Buckling 137, 139, 158 Burst 129 Butterfly valve 25, 165 Bypass 77, 82, 164 Cantilever 157
Coating, bitumen 191 coal tar 191 epoxy 191 glass fibre 191 mortar 118 tape 191 Cohesion 172, 174 Collapse 132 Collar
139
Concrete pipes 114, 163 prestressed 116, 163 properties 122 Compound pipes 27, 3 4 Compressibility 15 Computers, analogue 32 digital 34, 60, 65 mini 179 Conductivity 193, 197 Conductor 179, 193 Cone valve 167 Consolidation 135 Constraint 41
Capacity factor 4
Container 2
Cash flow 8
Contraction coefficient 177 Convection 196 Conversion factors 224
2 30
Core 116, 119
Entrance l o s s 25
Corrosion, galvanic 192
Epoxy 191
Cost, pipe 3
Equivalent diameter 27, 51
power 4,11 pumping 4, 9
length 27 Escher Wyss 145
Counterbalance 168
Excavation 181
Coupling 188
External load 101, 132
Crack 114, 117 Creep 122, 124, 164
Fabricated bend 175
Cross section 221
Factor of safety 115
Crotch plate 145
Flange 189, 208
Crown 119, 136, 149 Crushing 123 Current, electric 193 impressed 194 stray 196 Cylinder 128
blank 190 puddle 190 Flexible pipe,128, 132 Floating 185 Fittings 162, 221 Flow measurement: bend meter 177
Deflection 136,155,138,145
electromagnetic 178
Design factor 128
mass 178
Diameter 8
mechanical 177
Discharge coefficient 176
nozzle 176
tank 71
orifice 170, 176
Discount rate 10
venturi 175
Dissolve 92
volume 178
Drain 182
Flow reversal 6 4 , 81
Dynamic programing 34
Fluidizer 186
thrust 171
Flywheel 68 Friction 16, 62, 106
Economics 2
Froude number 96
Elasticity 59 Electrode 193
Galvanic action 192
Electrochemical equivalent 193
Gamma ray 187
Electrolyte 192, 193
Gasket 189
Electromagnetic induction 178
Gate, valve 165
Ellipse 223
Globe valve 166
Embankment 105
Gradient, hydraulic 35, 96, 181
Emissivity 196
Graphical analysis 13, 60
Empirical 17 Energy 15
Haunch 119, 132, 183
2 3 1
Head l o s s 16, 176, 209 static 60
Linear programming 40, 49 Lining, bitumen 191
velocity 15, 117
coal tar 191
water hamer 58
epoxy 192
Heading 183
mortar 192
Heat loss 196
Load coefficient 103
History 1
soil 101
Hole 145
superimposed 108
Holiday detector 191
test 114
Hydraulic gradient 16,35,96,169,181
Longitudinal stress 120,129,156
Hydraulic jump 95
Loop 28,39,47
Loss coefficient 24 Impact factor 109
Losses 16, 24
Impeller 206 Industrial pipework 157 Inertia 66,
120
Mass, conservation o f 15 Materials 162
Influence factor 110, 111
Mechanical meter 177
Insulation, thermal 196
Membrane theory 149
Interest 9
Meter, bend 177
Isothermal 78
mechanical 177 nozzle 176
Joint, buttwelded 187
orifice 175
clampon 188
roto 178
factor 128
vane 178
flanged 189 screwed 187 sleeve welded 187 spigot and socket 117, 187 underwater 186 Kinetic energy 65, 81
venturi 175 Model 147, 149 Modulus of elasticity 124,135,219 Momentum 15,81 Motor 210 Needle valve 83, 167
Lagging 198
Network 27
Laminar flow 19
Node 28, 29
Lateral support 134
Nonlinear programming 50
Laying 181
Normal thrust 171
Lime 191, 193
Nozzle 176
Line load 110
Objective function 41 Ohm 193 Oil 1
2 3 2
Operating factor 4
Pump ,blockage
Optimization 33
booster,2,8,28,35
Orifice 71, 77, 170
centrifuga1,66,202 inertia 66
Partly full pipe 88, 95 Passive soil pressure 172 Pavement
111
power 4,11 types 201 Pumping costs 4,11
Perimeter, wetted 89 Pipe, asbestos cement,163,215,225
Radial stress 128, 160
cast iron,162,216,218
Radiation 196
concrete,114,163,213,215
Radio 179
plastic,163,164,218,219
Radius of stiffness
polyethylene,l63,164
Rail 1
PVC,163,164,216
Reducer 165
111
properties,225
Reflux valve 66,69,73,81,168
salt glazed clayware,211,213
Reinforced concrete 1, 114, 115
stee1,162,204,211,212
Reinforcing 114, 129, 145
UPVC,163,218
Relaxation of stress 117
Pipework,industrial 157
Release valve 83
Planning 1
Reservoir 13
Plastic 163, 183
Reticulation network 17, 27
Point load 109
Reynold's number 19
Poisson's ratio 129, 225
Ribbed pipe 164
Polarization 193 Polyethylene 163, 225 Potential energy 81 Power 210
pavement 1 1 1 pipe 114 Ring girder
159
joint 186
Present value 9
load 132
Pressure, externa1,101,119,139
stiffening 139
internal,l28
stress 128
over, 53,129
tension 129
water hammer,53
Road I
Prestressed concrete pipe 116
Roughness 20
Proctor density 109,139,183
Route 181
Profile, pipeline 80, 181 Protection,cathodic 192 water hammer 64 Puddle flange 190
Sacrificial anode 193 Saddle 158
2 3 3
Salt glazed 215, 217 Sand 20,
173,
174
Strutting 134 Sulzer 145
Scale 4
Superimposed load 108
Scouring 165
Support 114, 119
Screwed joint 187
Surface load 108
Sea 185
Surge 53, 70
Secondary stress 145, 157
Symbols 226
Section modulus 157
Systems analysis 8, 33
Services 181 Settlement 106
Telemetry 178
Shape 221
Temperature 159, 196
Shrinkage 122, 124
Tension rings 129
Sinking fund 9
Testing 181
Sluice valves 165,215,218
Thermal insulation 196
Snake 182
Thermoplastic 164
Socket joint 117, 187
Thickness 128,138,147,156,196
Soil load 101
Throttling 167
mechanics 101 properties 174, 183 Span 157
Thrust block 171 bore 183 Tie bolt 188
Specific mass 224
Torque 211
Specific speed 204
Towing 186
Spherical valve 168
Traffic load 109
Spindle 165
Transient pressures 53
Standards 208
Transport and Roads Research Lab 137
Starters 210
Transportation, air 1
Steady flow 15, ,58
programing 37
Steel.high tensile 123,162
rail 1
pipe 128,162,223
road 1
prestressing 122
waterway 1
properties 122 Steepest path technique 44 Stiffening rings 129 Stray current electrolysis 195 Strength,laboratory 114 Stress, bending 156,157
Trench 101, 181 Turbine 70 Uncertainty 10 Underwater laying 185 Uniform load 110 Unplasticized polyvinylchloride 163.225
Stress,circumferential 118,129,136,145 Stress, longituduanal 120,129,156 temperature 159
Vacuum 101, 139
234
Valve, a i r r e l e a s e 77, 96, 170 a i r v e n t 96, 169 B u t t e r f l y 25, 166 cone 167 c o n t r o l 167 g l o b e 25, 166 n e e d l e 25,83,167 r e f l u x 25,66,69,73,81,168 r e l e a s e 83,96 s p h e r i c a l 168 sleeve 83,167
s l u i c e 25,165,213,218 Vane,guide 145 V a p o r i z a t i o n 64,77,80,88,129 V e l o c i t y 15,18,148,176 V e n t u r i meter 175 V i c t a u l i c c o u p l i n g 189 Viking Johnson c o u p l i n g 188 V i s c o s i t y 1 8 , 21 Volute 202 Vortex 81,170 Wall 128 Water hammer 53 p r o p e r t i e s 224 s u p p l y 13 Wave 59 Web 145 Welding 129, 187 Wrapping 182, 190 Xray
187
Yield stress 139, 162