Attention is to be paid to the e GERMAN ATV-DVWK RULES AND STANDARDS

ATV-DVWK STANDARD ATV-DVWK-A 110E

Hydraulic Dimensioning and Performance Verification of Sewers and Drains September 2001 ISBN 3-936514-30-5

Distribution: GFA Publishing Company of ATV-DVWK Water, Wastewater and Waste

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E-mail: vertrieb@gfa-verlag.de • Internet: http://www.gfa-verlag.de

ATV-DVWK-A 110E

September 2001 2

Die Deutsche Bibliothek [The German Library]– CIP-Einheitsaufnahme

ATV-DVWK, German Association for Water, Wastewater and Waste: ATV-DVWK Set of Rules and /ATV-DVWK, Water, Wastewater, Waste.- Hennef: GFA Publishing Company of ATV-DVWK. Previously under the title: German Association for Water Pollution Control (ATV): ATV Set of Rules and Standards Standard A 110E. Hydraulic Dimensioning and Performance Verification of Sewers and Drains - 2001 ISBN 3-934984-03-7

All rights, in particular those of translation into other languages, are reserved. No part of this Standard may be reproduced in any form - by photocopy, microfilm or any other process - or transferred into a language usable in machines, in particular data processing machines, without the written approval of the publisher.

Publisher: ATV-DVWK German Association for Water, Wastewater and Waste, Theodor-Heuss-Allee 17, D-53773 Hennef, Germany

Marketing: GFA - Publishing Company of ATV-DVWK, Hennef

Original German version set and printed by: DCM, Meckenheim, Germany

GFA Gesellschaft zur Förderung der Abwassertechnik e.V. Hennef 2001

ATV-DVWK-A 110E

September 2001 3

Preamble to the August 1988 Edition The revision of ATV Standard A 110 had to give consideration to a host of suggestions from practice, to realise the firm ideas of the ATV Advisory Board and to deal with fundamental relationships. The questions relating to the recalculation of existing sewer networks (performance verification), in particular, made it necessary to expand the theoretical part of the draft of the new version to a greater extent than is customary in a standard, which is not meant to be a manual.

Consequently, the Working Group first devised a comprehensive draft - the October 1985 version - which contained more information, explanations and source material than was necessary for this Standard. The final print is an abbreviated version which is essentially limited to describing the procedure in individual practical cases and no longer contains any explanatory information or justifications.

The following accompanying publications to this ATV Standard appeared in Vol. 1/1989 of the Korrespondenz Abwasser:

Howe, H.O.: Grundzüge des neuen ATV-Arbeitsblattes A 110 [Main features of the new ATV Standard A 110.]

Haendel, H.: Auswirkungen der Neufassung des ATV-Arbeitsblattes A 110 auf die hydraulische Nachrechnung von Abwasserkanälen [Effects of the new edition of ATV Standard A 110 on the hydraulic recalculation of sewers].

Knauf. D.: Fließtiefen und Durchflüsse für gegliederte Querschnitte [Flow depths and throughflows for structured cross-sections].

Ueker, K.J.: Abfußberechnung in Abwasserkanälen unter Berücksichtigung seitlicher Zuflüsse [Flow calculation in sewers taking into account lateral inflows].

Unger, P.: Grundlagen der kb-wert-Festlegung im Arbeitsblatt A 110 [Bases of the determination of the kb value in Standard A110].

Preamble to the present Edition Due to suggestions received, amendments and corrections which have become necessary and due to having to take into account the European Standard DIN EN 752 “Drain and sewer systems outside buildings”, Part 4 “Hydraulic design and environmental considerations”, it has become necessary to revise the August 1988 edition of ATV A 110.

ATV Standard ATV-A 110 is, within the framework of European standardisation, a source for additional information in accordance with DIN EN 752, Part 4, Section 4, and thus a component part of this European Standard.

The results of various ATV research projects have flown into the revised edition. Fortunately the following were available for this:

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- Beeinflussung der Leistungsfähigkeit von Kanalstrecken durch konstruktive Veränderungen im Bereich der Schächte [Influencing of the efficiency of sewer reaches through design changes in the area of shafts] (ATV/Deutsche Bundesstiftung Umwelt [German Federal Foundation of the Environment]

- SIMK - Simulation von Teilfüllungenskurven [Simulation of partially filled curves] (ATV 25/97 and ATV 31/99)

- Schießender Abfluß in Krümmerschächten [Supercritical outflow in shafts for change of flow direction] (ATV 09/98).

Important amendments in comparison with the August 1988 edition are:

- separation of the procedure with dimensioning and with performance verification - revised, expanded representation of flow losses in shafts with and without

impounding - new type of treatment of flow in shaft structures with supercritical discharge with

simultaneous flow diversion - analytical treatment of flow in noncircular profiles and with partial filling conditions - dropping taking into account the form coefficient f - generalisation of the treatment of discontinuous flow by the introduction of the

factor m - calculation of the energy conversion in the outlet area of steep sections - expanded new version of the treatment of flat reaches and deposits - revision of the calculation for open channels and structured cross-sections. With the calculation of surface curves the uninterrupted transfer from partial filling via complete filling to pressure discharge (impounding) also taking into account the impounding of shafts, is possible. Simplified mathematical models require model-specific approaches, which would lie outside the scope of ATV Standard A 110.

For reasons of easier reading and comparability with the August 1988 edition of ATV-A 110, the use of new [German] spelling rules introduced in some Federal German States has been dispensed with.

Dr.-Ing H. O. Howe, Chairman

ATV-DVWK-A 110E

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Authors

This Standard has been revised by the ATV-DVWK Working Group "Hydraulic Calculation of Sewers and Drains” within the ATV-DVWK Specialist Committee “Planning of Drainage Systems”. The Working Group has/had the following members: Prof. Dr.-Ing. E. Billmeier, Köln Dr.-Ing. P. Drewniok, Leipzig Dr.-Ing. N. Engel, Berlin (from 1998) Dipl.-Ing. K-H. Flick, Köln (Chairman from 10/2000) Dipl.-Ing. H. Haendel, München, r.i.p.s. Prof. Dr. sc. techn. W. H. Hager, Zürich Dr.-Ing. Harald O. Howe, Köln (Chairman to 9/2000) BD Dr.-Ing. H. Krier, Frankfurt Dipl.-Ing. G. Milkov, Hamburg Obering. H. Schmidt, Erkrath Dipl.-Ing. F. Schweinebraten, Kassel (from 1996) Prof. Dr.-Ing. W. Tiedt, Darmstadt Dr.-Ing. P. Unger, Lich Prof Dr.-Ing. F. Valentin, München Dipl.-Ing. D. Wengler, Rheinfelden (from 1996) Dipl.-Ing. G. Zanker, München (to 1996)

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Contents Page

Preamble to the August 1988 edition 3

Preamble to the present edition 3

Authors 5

Notes for users 8

1 Scope 8

2 Hydraulic principles 8

2.1 Forms of discharge and discharge processes 8 2.2 Defining equations 9 2.3 Drag coefficient 12 2.4 Loading assumptions and limiting values 15

3 Formulas for calculation in the standard case 15 3.1 Closed channels 15 3.1.1 Complete filling 15 3.1.2 Partial filling 16 3.1.3 Special profiles with dry weather channel 20 3.2 Open channel 20 3.2.1 Calculation of flow depths 20 3.2.1.1 Normal water depth hn 20 3.2.1.2 Critical depth hcrit 21 3.2.1.3 General flow depth h(x) 21 3.2.2 Calculation of flows 22 3.2.2.1 Normal discharge Qn 22 3.2.2.2 Critical discharge Qcrit 22 3.2.2.3 Discharge curve Q = f(h) 23 3.2.3 Flow depths and flows for structured cross-sections 24

4 Dimensioning and performance verification 26

4.1 Determination of roughness 26 4.1.1 General measure of roughness k 26 4.1.2 Operational roughness kb 27 4.2 Calculation of individual losses 28 4.2.1 Loss coefficients as a result of positional inaccuracies and changes (ζPi) 29 4.2.2 Loss coefficients for pipe connections (ζPC) 29 4.2.3 Loss coefficients at inlet fixtures (ζin) 29 4.2.4 Loss coefficients for control shafts (ζC) 30 4.2.5 Loss coefficients for special shafts (ζS) 32 4.2.6 Loss coefficients (ζfd) and verifications for flow diversion 33 4.2.6.1 Subcritical flow 34 4.2.6.2 Supercritical flow 34 4.2.7 Loss coefficients for combining structures (ζCS) and verification for flow merging 36

4.2.7.1 Subcritical discharge 36 4.2.7.2 Supercritical discharge 38

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4.2.8 Other loss coefficients 38 4.3 Dimensioning 38 4.4 Performance verification 39

5 Flow with lateral inflow (discontinuous flow) 40

5.1 Effect of lateral inflow 41 5.2 Simplified procedure 41 5.2.1 Dimensioning (selection of a constant replacement flow) 41 5.2.2 Performance verification 42

6 Flat stretches and depositing 42

7 Steep stretches and air transfer 45

7.1 Steep stretches - inflow 46 7.2 Steep stretches - throughflow 46 7.3 Steep stretches - outflow 47

8 Special structures 49

9 Pressure and vacuum drainage, compressed air flushed wastewater transport pipelines, wastewater pumping stations with pressure pipelines 50

10 Private property drainage 51

11 Cost aspects 52

12 Literature 53

13 Symbols, units and definitions 56

Appendix 59

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Notes for Users

This ATV Standard is the result of honorary, technical-scientific/economic collaboration which has been achieved in accordance with the principles applicable for this activity (statutes, rules of procedure of the ATV and ATV Standard ATV-A 400). For this, according to precedents, there exists an actual presumption that it is textually and technically correct and also generally recognised.

Everyone is at liberty to apply this Standard. However, an obligation for application can arise from legal or administrative regulations, a contract or other legal reason.

This Standard is an important, however, not the sole source of information for correct solutions. With its application no one avoids responsibility for his own action or for the correct application in specific cases; this applies in particular for the correct handling of the margins described in the Standard.

1 Scope

ATV Standard A 110 “Guidelines for the hydraulic calculation of sewers” in its original version (October 1965) [not translated into English] had presented the calculation of the discharge capacity and flow rate on theoretically better assured bases. Building on the results from Prandtl and Colebrook, the discharge condition, which is based on a dimensioning, can be better described. Furthermore, the ever increasingly important subsequent calculations of existing networks, without application of the bases, would contain considerable uncertainties.

The first revision (August 1988) updated this Standard ATV-A 110 [translated into English as ATV Standard ATV-A 110E] as “Standards for the Hydraulic Dimensioning and Performance Verification of Sewers and Drains”. This second revision builds further on this.

A summary of the rules due to European standardisation and their adoption through DIN EN 752-4 serves for current orientation. The symbols and terms used are - as far as possible - in agreement with DIN 4044 “Hydromechanics in hydraulics; definitions”, DIN 4045 “Wastewater engineering; vocabulary” and DIN EN 752 “Drain and sewer systems outside buildings”, Part1 “Generalities and definitions”.

2 Hydraulic Principles 2.1 Forms of Discharge and Discharge Processes

The discharge in sewers and drains is characterised by a great number of simultaneously possible forms of discharge, the most important of which in the area of sewer systems are compiled in Table 1 below. All details refer to parameters in the direction of flow.

Under the given prerequisites and ignoring further terms of the complete differential equation there result gradually simplified calculation statements, which are put together in Table 2 with their associated designations. The equations in Line (0) describe in detail the discharge process in sewers and drains in a generally valid form. Line (1) applies for transport pipelines without lateral in- or outflows along the calculated stretch.

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Further simplifications to Line (2) to Line (6) result through random omission of individual influencing elements. Neglecting the elements

xv

gvand

tv

g1

∂∂

⋅∂∂

⋅

errors occur which can have the opposite tendency.

For transport pipelines the application of the equations in Line (4) should therefore definitely be given preference over the calculation methods according to Line (2) or Line (3). A more extensive assessment of the correctness of the various simplified assumptions has to date not been possible, instead one is reliant on control calculations in individual cases.

The necessary numerical assumptions (increments ∆x, ∆t; their ratio; convergence criteria) are to be observed for solutions based on the complete statements in Lines (0) or (1). Simplification according to Line (7) is applied for the calculation of partial filling conditions (see Sect. 3.1.2), but describes a discharge form which - with the exception of very long transport pipelines - does not occur in the practical operation of sewers and drains. Nevertheless the use of this relationship, represents a useful estimation on whose accuracy no exaggerated demands should be placed and whose limits are redefined in this standard.

2.2 Defining Equations

The defining equation for the mean flow rate is

AQv = (1)

In which the following apply

v the mean velocity in the direction of flow [m/s] Q the flow also known as discharge or volume flow1) [m3/s] A The flow cross-section, in the case of the completely filled circular pipe [m2]:

with4dA

2⋅π=

d as the actual internal diameter of the circular pipe (clear width) [m].

The resistance formula2) for the friction losses, taken as being evenly distributed along the flow stretch in the direction of flow, is

g2v

r4lh

2

hyf ⋅⋅λ= (2)

In which the following apply: __________________ 1) In wastewater engineering the flow is usually expressed in l/s; for general considerations m3/s is selected for reasons of

dimensioning. 2) This resistance formula was developed by D’Aubuisson de Voisin (1834) and Weisbach (1845); it is often erroneously attributed

to Darcy and also incorrectly referred to as the “Darcy-Wiesbach Equation”

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hf energy fall

λ a non-dimensional resistance coefficient [1]

rhy the hydraulic radius [m], defined as

)fillinghalf/completewithpipecircularthefor(4d

lperimeterwettedAtionseccrossflowr

Phy =

−=

l the length [m] of the drain or sewer section.

Converted Eqn. (2):

g2v

h41

lhJ

2

ry

fE ⋅⋅λ== (3)

Here, JE is the designation for the gradient of the energy curve.

In the case of steady-state, uniform discharge (Line 7 of Table 2), the so-called normal discharge, the gradient of the energy curve JE is equal to the sole gradient Jso. This can, in general, be formed with the projection length of the pipeline. Up to a gradient of some 200 ‰ the inherent error is less than 2 %. On the other hand, with steep stretches, the actual pipeline length is to be applied.

Table 1: Forms of discharge in sewers and drains

Designation Criteria Designation Criteria

steady-state 0t()

=∂∂ unsteady 0

t()

≠∂∂

uniform 0x()

=∂∂ non-uniform 0

t()

≠∂∂

continuous q = 0 discontinuous q ≠ 0 subcritical Fr < 1 supercritical Fr > 1 laminar Re < 2320 turbulent Re > 2320 single-phase wastewater multi-phase wastewater + air

With the hydraulic dimensioning and calculation tasks within wastewater engineering, there are not only continuously active influences such as wall friction to be taken into account but also the effects of locally occurring individual losses to be included in the calculation.

In this the following apply (assuming one-dimensional streamtube theory):

t()

∂∂ the variability of a flow value (), e.g. Q, h, v with time t

x()

∂∂ the variability of a flow value (), e.g. Q, h, v with flow path x

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q the lateral inflow per unit of length in the direction of flow [m3/(s ⋅ m)] Fr Froude Number [1] Re Reynolds Number [1]

Table 2: Calculation formulations for flow processes in sewers and drains

Line Type of motion Motion equation Continuity equation

0 unsteady non-uniform discontinuous

FSo JJxh

Agqvm

xv

gv

tv

g1

−=∂∂

+⋅⋅

⋅+∂∂

⋅+∂∂

⋅ qtA

xQ

=∂∂

+∂∂

1 unsteady non-uniform FSo JJ

xh

xv

gv

tv

g1

−=∂∂

+∂∂

⋅+∂∂

⋅ 0tA

xQ

=∂∂

+∂∂

2 unsteady simplified non-uniform

FSo JJxh

tv

g1

−=∂∂

+∂∂

⋅ 0tA

xQ

=∂∂

+∂∂

3 simplified unsteady non-uniform FSo JJ

xh

xv

gv

−=∂∂

+∂∂

⋅

0tA

xQ

=∂∂

+∂∂

4 simplified unsteady simplified non-uniform FSo JJ

xh

−=∂∂ 0

tA

xQ

=∂∂

+∂∂

5 steady-state non-uniform FSo JJ

xh

xv

gv

−=∂∂

+∂∂

⋅

0xQ

=∂∂

6 steady-state simplified non-uniform FSo JJ

xh

−=∂∂ 0

xQ

=∂∂

7 steady-state uniform (normal discharge)

0xh

=∂∂ 0

xQ

=∂∂

In which the following apply Q throughflow [m3/s] q lateral inflow per unit of length in the

direction of flow (assumed steady-state) [m3/(s ⋅ m)]

A flow cross-section perpendicular to the sole [m2]

Jso sole gradient (with open channel possibly not constant) [1]

JF friction gradient3) [1]

x path co-ordinate in direction of flow [m] t time co-ordinate [s] h filling height in profile or depth of water (perpendicular to

sole) or the pressure head in completely filled drains at the sole of the pipe or profile [m]

v mean velocity [m/s] in a cross-section in direction of flow g acceleration due to gravity [m/s2] m factor with inclusion of additional losses [1] (see also Eqn.

52)

In the general, comprehensive form which includes these individual losses, Eqn. 2 becomes:

g2v)

r4l(h

2

hyf ⋅ςΣ+⋅λ= (4a)

_________________ 3) The friction gradient JF, with sufficient accuracy, may be replaced by the energy gradient JE (see also Eqn. 3)

ATV-DVWK-A 110E

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or for the circular pipe

g2v)

dl(h

2

f ⋅ςΣ+⋅λ= (4b)

Here ζ designates a non-dimensional coefficient (see Sect. 4), through which the size of the individual losses hf,i, referred to the velocity head v/2g, is determined

g2vh

2

i,f ⋅ς= (4c)

With the inclusion of Eqn. 1 one obtains the general flow formula

Ehyhy

Jr4g2

lr41AvAQ ⋅⋅⋅

ζΣ⋅+λ

⋅=⋅= (5a)

and thus for the circular pipe

E

2

Jdg2

ld1

4dvAQ ⋅⋅⋅

ζΣ⋅+λ⋅

⋅π=⋅= (5b)

For simplified handling, the combination of the various loss coefficients into one operational drag coefficient λb is proposed. This is defined as

ζΣ⋅+λ=λlr4 hy

b (6)

With this Eqns. 5a and 5b become

Ehyb

Jr4g21AQ ⋅⋅⋅λ

⋅= (7a)

and thus for the circular pipe

Eb

2

Jdg214dQ ⋅⋅⋅

λ⋅

⋅π= (7b)

2.3 Drag Coefficient

In connection with the resistance coefficient, a distinction is made between the following cases and ranges for turbulent flow (laws of resistance):

1) Ideal or hydraulically smooth behaviour (smooth curve) according to Prandtl:

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λ⋅⋅−=−λ⋅⋅=

λ Re51.2lg28.0)lg(Re21 (8)

2) Completely or hydraulically rough behaviour (completely rough area) according to Prandtl

⋅⋅−=+

⋅=

λ hy

hy

r4k

71.31lg214.1

kr4

lg21 (9)

3) Technically rough behaviour (transitional area) according to Colebrook

⋅+

λ⋅⋅−=

λ hyr4k

71.31

Re51.2lg21 (10)

Only Eqn. 10 is of practical importance for use in sewers and drains. It is generally designated as the Prandtl-Colebrook equation.

Here the following apply: λ drag coefficient [1]

Re Reynolds Number

ν

⋅= hyr4v

[1]

k the hydraulically effective roughness of the internal pipe wall, defined by the Prandtl-Colebrook Equation and to be determined precisely through hydraulic trials only [mm; m].

rhy the hydraulic radius, calculated from the clear dimensions of the flow - in the case of circular pipes it is 4rhy = d [m].cross-section.

In the so-called Moody diagram, Fig. 1, the relationships expressed in the above formulas are represented in graphic form. A detail of the part of this diagram interesting for the most frequent cases in practice is contained in Appendix A5.

The application of the relationships according to Eqns 8, 9 and 10 is permitted for both circular cross-sections and non-circular cross-sections and that is for closed profiles and also open channels. Even in the case of cross-sections, which are far from circular, the taking into account of a form coefficient f is dispensed with for the correction of the hydraulic radius rhy

4). The kinematic viscosity is defined as

]s/m[v 2

ρη

= (11)

Here the following apply: η dynamic viscosity [kg/(m ⋅ s)] ρ density [kg/m3] The dependence of kinematic viscosity on the temperature is shown in Table 3.

_________________ 4) For this see also the report of ATV Research Projects 25/97 and 31/99

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Fig. 1: Moody diagram for completely filled circular pipes

Table 3: Kinematic viscosity for various temperatures (values for pure water)

T [°C] 5 10 15 20 25 30 ν ⋅ 106

s

m2

1.52 1.31 1.15 1.01 0.90 0.80

With the calculation of sewers and drains as a rule the following is set:

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September 2001 15

⋅=ν −

sm1031.1

26

Within this the normally higher temperatures and the other composition of wastewater compared with pure water are taken into account.

2.4 Loading Assumptions and Critical Values

The load assumptions for determining the flow and water surface profiles can be found in ATV Standard ATV-A 118E. Information concerning the surface run-off (e.g. percolation, pit losses and flow times) is given in ATV-A 118E and ATV-M 165.

DIN 1986, Part 2 applies together with DIN EN 12056 for private property drains within buildings; outside of buildings the standard series DIN EN 752 applies.

With the dimensioning of sewers and drains, for reasons of correct operation, attention is to be paid to the observance of critical values for flow velocities. Flat stretches (see Sect. 6) are characterised by operation in the vicinity of the minimum velocity, steep stretches (see Sect. 7) by operation with relatively high velocities.

Global critical values for both ranges are no longer suitable. The recommended procedure is dealt with in detail in Sects. 6 and 7.

3 Formulas for Calculation in the Standard Case

3.1 Closed Channels

The dimensioning of new construction and renovation of sewers and drains is normally oriented towards complete filling (Index: V), whereby this should not be exploited to the full. If the dimensioning discharge reaches 90 % of the discharge capacity Qv, it is recommended to select the next largest cross-section. This gives global consideration to the following:

- undercutting of nominal dimensions within the permissible framework in accordance with DIN 4263, Sect. 2.1,

- reductions of cross-section due to unavoidable deposits of up to 3 % of the total cross-sectional area even if sewers and drains are maintained regularly,

- equation of the actual sewer length l with its projection l’.

With performance verification of sewers and drains the respective permitted back-up heights or the overdamming frequency are relevant (see DIN EN 752-2). Procedure for dimensioning is presented in Sect. 4 of this standard.

3.1.1 Complete Filling

For circular profiles the general discharge formula is:

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⋅⋅⋅

⋅+

⋅⋅⋅ν⋅

⋅−⋅⋅π

= EE

2

Jdg2d71.3

kJdg2d

51.2lg24dQ (12)

For non-circular profiles the general discharge formula is:

⋅⋅⋅

⋅+

⋅⋅⋅ν⋅

⋅−⋅= EhyhyEhyhy

Jr4g2r84.14

kJr4g2r4

51.2lg2AQ (13)

A mathematical consideration of the deviation of any profile (form coefficient f) from the circular profile is dispensed with, as for all normal cross-section shapes in sewer systems, f can be set = 1.

Eqn. 13 applies for all profile shapes used in sewer systems such as, for example, oval, tapering and channel cross-sections, also with considerable deviation from a circular shape.

In the case of steady-state, uniform discharge in pipes filled to the top, Eqns. 12 and 13 the energy gradient JE is replaced by the sole slope JSo.

In keeping with the details given in Sect. 4.1, the roughness k can be replaced by the operational roughness kb. In all cases - circular profile as well as non-circular profile - DIN 4263, Sect. 3.2 (clear width) is valid without restriction and must be observed5).

3.1.2 Partial Filling

Simplified assumptions can be made for the calculation of discharge processes with partial filling. For steady-state, uniform discharge (normal discharge) the energy gradient and water surface profile run parallel to the sole (Fig.2). With partial filling curves the partial filling values are referred to complete filling.

Water surface profile calculations are absolutely essential with non-uniform discharge (see also Sect. A3.)

Degree of filling h/H and the therefrom calculated geometrical parameters (see Table 14) are non-dimensional; with circular pipes H = d. The profile H is always measured perpendicular to the pipe axis The filling height h must therefore also be measured perpendicular to the pipe axis. The filling height is thus not the same as the depth of water h’, but rather linked with this parameter h’ by the relationship

'hcos/h =φ (14)

and therefore with cleaned sewers is always somewhat smaller than the depth of water6).

____________________ 5) Clear widths are the actual dimensions of the cross-sections. The dimension of the clear width should correspond with the

characteristic value of the nominal width. For the purpose of hydraulic dimensioning in wastewater engineering and in general hydraulic engineering may only be assumed to be equal to the nominal width if the undercutting of the cross-section, referred to the numerical value of the nominal width, does not exceed 5 %. In this case the average clear diameter undercuts the characteristic value of the nominal width by ca. 2.5 %.

6) With gradient values ≤ 200 ‰ (0.200; 1:5) h = h’ and l = l’ can be set; the resultant error is ≤ 2 %. Eqns 14 and 15 are to be used for calculation with larger sole gradients

For the length the following applies

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September 2001 17

φ= cos/'ll (15)

For the calculation of partial filling conditions in all forms of cross-section the form coefficient f = 1 is also applied.

Fig. 2: Normal discharge with partially filled sewers

The partial filling curves - see also 7)8)9) - the following applies for flow velocities

625.0

V,hy

T,hy

V

T

rr

vv

= (16)

and for discharges

625.0

V,hy

T,hy

V

T

V

T

rr

AA

QQ

⋅= (17)

_________________ 7) Franke, P.: Die Rauhigkeitsverhältnisse im teilgefüllten Rohr [Roughness conditions in the partially filled pipe]. 8) Tiedt, W.: Hydrodynamische Untersuchungen des Teilfüllungsproblems [Hydrodynamic investigations of the partial filling problem]. 9) Sauerbrey, M.: Abfluß in Entwässerungsleitungen unter besonderer Berücksichtigung der Fließvorgänge in teilgefüllten Rohren

[Discharge in drainage pipelines under special consideration of flow procedures in partially filled pipes].

Investigations have shown that the unrestricted application of Eqns. 16 and 17 is justified for all forms of cross-section10). In addition to this, the possibility is pointed out of

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September 2001 18

calculating the normal depth of water in partially filled cross-sections directly using iterative evaluation of Eqn. 13.

The theoretical investigations by Tiedt8) have confirmed the practical experiments by Sauerbrey9) insofar as the influence of air friction on the discharge behaviour of partially filled, closed cross-sections can be neglected. The partial filling curves thus have a reverse bending part with a discharge maximum with partial filling, to which the greatest possible stable normal depth of water is to be assigned.

Due to the problems of aeration or air entrainment in sewer pipelines, combined with the resultant risk of the surcharging of the pipelines, the partial filling curves for discharges are terminated at

0.1QQ

V

T =

With profiles with a flat roof it is recommended that the partial filling curves for discharges are terminated, dependent on the width of the cross-sections, 10 to 20 cm below the crown, whereby the following is possible in these cases:

0.1QQ

V

T >

Fig.3: Partial filling curves for circular, oval and tapering profiles

__________________ 10) ATV: SIMK - Simulation von Teilfüllungskurven,. Abschlußbericht des Forschungsvorhabens (ATV 25/97 and ATV 31/99)

[Simulation of partial filling curves, final report of the research project (ATV 25/97 and ATV 31/99)].

In Fig. 3 is shown the plot of the partial filling curves for the forms shown in Fig. 4 (oval profile) and Fig. 5 (tapering profile), whereby the curve for the oval profile lies above the

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middle curve for the circular profile (Fig. 3) and below it for the tapering cross-section. Profiles with similar hydraulic behaviour can be grouped together for the practical calculation of the partial filling values (for this see also Appendix A2).

The partial filling curves for the flow velocities are also terminated and that is at the point hT/d at which QT/QV = 1.0 is reached. For profiles with flat roofs this also takes place at 10 to 20 cm below the crown.

Fig. 4: Oval profile with a) full filling and b) partial filling

Fig. 5: Tapering profile with a) full filling and b) partial filling

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3.1.3 Special Profiles with Dry Weather Channel

The calculations with complete filling for this type of profile are carried out using Eqn. 13 over the complete cross-section, without hydraulic structuring in the partial profile.

The calculation for partial filling is in accordance with Sect. 3.2.3.

3.2 Open Channel

Sewers in the form of open channels pose questions with the hydraulic dimensioning as well as with later performance verification, which in part deviate from the treatment of closed profiles. Nevertheless, it should be noted that, in the case of the calculation of water surface profiles, the conditions in partially filled, closed profiles correspond with those, which are dealt with here for open channels. In this respect the statements of this section are also valid for general solutions for sewers and drains in the operational state of partial filling.

Under normal conditions of operating open, natural channels the Reynolds No. is so high that, in the sense of the definition in accordance with Eqn. 9, one has, as a rule, to reckon with flow in the completely rough range.

Therefore it is possible in this case also to apply other relationships for the determination of the energy gradient. As an alternative to the treatment according to Prandt-Colebrook the relationships according to Manning-Strickler can be recommended here. For the flow formula these are

2/1E

3/2hySt Jrkv ⋅⋅= (18)

and for the discharge formula

2/1E

3/2hySt JrkAQ ⋅⋅⋅= (19)

using the coefficient

s

mk3/1

St in accordance with Manning-Strickler,

which, in the fully rough range, is dependent on wall roughness only (see Appendix A7).

3.2.1 Calculation of Flow Depths

3.2.1.1 Normal Water Depth hn

In special cases, for example in the case of long, disturbance-free flow sections, normal discharge prevails. Under this term are gathered those discharge processes with which, in the flow direction x, none of the flow parameters involved changes or with which the assumption of this idealised flow condition is necessary for reasons of convenience (estimation, analytical simplification). Under these circumstances the friction gradient JF, the energy gradient JE, the water profile gradient JW and the sole gradient JSo are, mathematically, equal to each other: Thus the following applies

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JF = JE = JW = JSo (20)

The depth of water with which the condition in accordance with Eqn. 20 is met, is called the normal water depth hn, whose calculation takes place with the aid of Eqns. 13 and 20 or 19 and 20.

If Q, JSo, d, kb and v are specified in Eqn. 13, or Q, KSt, and JSo, and the form functions A = f(h) and rhy = f(h) in Eqn. 19, then the flow depth sought, the normal water depth hn can be determined mathematically by iteration.

Whichever formula is selected for the calculation of the normal depth of water the associated discharge can result as sub-critical (Fr < 1), supercritical (FR > 1) or also in the transition condition (FR = 1) (see Table 1) Thus whether subcritical or supercritical normal discharge is present is decided by the size of the Froude number.

3.2.1.2 Critical Depth hcrit

Initial relationship for each dealing with flow characteristics in the boundary area, with velocity distribution assumed to be uniform, is the relationship

1Ag

bQFr 3

22 =

⋅⋅

= (21a)

where

Fr Froude Number [1] b water level width [m].

For partially filled circular sections the following approximation applies

4

22

hdgQFr

⋅⋅= (21b)

Attention is drawn to ATV Standard ATV-A 111 (Sect. 3.2) for oval and tapering profiles.

Warning is given on the simplified definition of the Froude Number hg/vFr ⋅= as h signifies the actual flow depth in the special case of the rectangular cross-section only.

The critical depth always appears with transition between subcritical and supercritical discharge. Within wastewater engineering it is to be observed with large differences in the sole gradient of neighbouring reaches (e.g. drops, change of cross-section). Places at which the critical depth appears mark a decoupling of parts of the wastewater network, which is to be noted when dealing with the subject mathematically.

3.2.1.3 General Flow Depth h(x)

Discharge takes place unevenly, unsteadily and discontinuously. The usual form of the approximation originates from Line 5 of Table 2 and uses the relationships for the unsteady, uneven discharge only, i.e. the water depth rises (backwater curve) or falls (decline curve) in the direction of flow with the result that neither the friction gradient nor

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the energy gradient nor, as a result of this, the water surface profile are known from the outset.

Under these conditions the flow depth sought h varies from place to place, that is h = f(x) and can only be found by integration of the differential equation of the water surface gradient. This results from the relationships relevant in the case under consideration in accordance with Line 5 of the tables is

2FSo

Fr1JJ

dxdh

−−

= (22)

whose integration delivers the desired result.

ax 2FSo hdx

Fr1JJ

)x(h +−

−= ∫ (23)

As both JF and Fr2 represent functions of h this also succeeds only by iteration. The initial water depth hi, to be accounted for as additive constant, is either specified (e.g. as design retention level) or is to be calculated from the flow parameters of designated positions, so-called monitoring cross-sections, for example in the form of a critical depth hl. If the discharge is supercritical (Fr > 1), integration is to be downstream.; if there is sub-critical discharge present in the integration zone (Fr < 1), then integration as a rule is to be taken against the direction of flow (upstream).

The calculation of flow resistance or energy head losses can be according to Prandtl-Colebrook or Manning-Strickler.

3.2.2 Calculation of Flows

3.2.2.1 Normal Discharge Qn

Normal discharge is the throughflow or volume flow in an open prismatic channel present with normal water depth.

In these cases, i.e. with uniform discharge with JF = JE = JW = JSo, the throughflow can be determined directly knowing the parameters JSO, kb, g, ν (applying the Prandtl-Colebrook equation) or JSo, kSt (applying the flow formula according to Manning-Strickler) as well the form functions A = f(h), rhy = f(h) and the flow depth h = hn from Eqn. 13 or Eqn. 19.

3.2.2.2 Critical Ql

The critical discharge Qcrit is that throughflow or volume flow which discharges in an open channel under local or continuous critical conditions, i.e. with Fr of Fr2 = 1.

If one resolves Eqn. 21a which is relevant for this, then one obtains the following conditional equation for the critical discharge in an open channel

bAgQ

3

crit⋅

= (24a)

and as approximation for partially filled circular cross-sections

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4crit hdgQ ⋅⋅= (24b)

3.2.2.3 Discharge Curve Q = f(h)

The graphic or numerical assignment of flow depths h or water levels hNN to throughflows Q at designated positions xi of open channels is designated as discharge (also rating or level) curve. In the special case of normal discharge requires only an evaluation of Eqn. 13 or Eqn. 19.

On the other hand, in the normal case, i.e. with non-uniform discharge, a water surface profile calculation is required in advance which, for these purposes, is to be repeated with the specification of various throughflows and which produces, dependent on the throughflow, different results for JF, Fr and, if required, hA.

If, however, flow depths and associated water level gradients known through on-the-spot measurements, then the discharge curve can be determined semi-experimentally and presented in a self-contained form. In the sense of sufficient accuracy such measurements should be carried out with steady discharge and as far as possible at a point which lies in the region of a decline curve for the water level profile.

The relevant flow formulas for this for the case of prismatic channels are given below:

a) for evaluation in accordance with the Prandtl-Colebrook equation

⋅⋅⋅

⋅+

⋅⋅⋅ν⋅

−⋅= EhyhyEhyhy

Jr4g2r84.14

kJr4g2r4

51.2lg2AQ (13)

b) for evaluation in accordance with the Manning-Strickler flow formula

2/1R

2/1hy

hy

3/1hy

2St

So Jr

Ag

rbdxdh

rk1

Jdx/dh1

AQ ⋅⋅

⋅

⋅⋅−

⋅

−⋅= (25)

with

⋅

⋅−⋅−= 3

2

SoF AgbQ1

dxdhJJ (26)

as well as h and dxdh (or ∆h/∆x), which are produced experimentally.

Note: Q must be calculated iteratively.

3.2.3 Flow Depths and Flows for Structured Cross-Sections

In structured channel profiles and in particular with small flow depths above the benching, different flow depths, different flow velocities and thus also secondary flows arise in the partial discharge areas.

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Through this the hydraulic efficiency reduces as the kinetic part of the energy reduces with the non-uniformity of the velocity distribution, and the friction or loss component increases through additional interactions between the flow channel and lateral flow cross-sections. The additional energy necessitated by the secondary flows is mathematically accounted for through the correction factor α with α > 1. With this water depths and discharges are to be determined only with involvement of the energy curve. A calculation of normal flow over the sole gradient is excluded.

With cross-sectional shapes with discontinuous increase of the wetted perimeter and with only small change to the water depth (as, for example, with profiles with dry weather channel and double-sided benching) the different flow velocities cannot be ignored.

The influence of non-uniform distribution of velocities can be neglected if the flow depth above the benching exceeds the depth of the channel (hB > HCh).

In the region of smaller flow depths over the lateral benching (hB ≤ HCH) the following simplified approximation procedure is recommended for the calculation of the hydraulic efficiency and the associated characteristic values of the flow conditions:

- separation of the structured cross-section in the partial flow region such that the variations in velocity in the individual partial cross-section are insignificant. In this case the delineation can be by imaginary, perpendicular interfaces.

- Increase of the geometrically wetted perimeter Ip,Ch of the flow channel through the correction factor ∆lp* to the hydraulically effective wetted perimeter. For the calculation of , with hB ≤ HCh, the following applies as approximation:

+

−=∆ChB

BBp Hh

h21h*l (27)

The following results from this:

R,p*

L,p*

Ch,pCh,p* llll ∆+∆+= (28)

- calculation of the water levels and low conditions which are to be set with specified discharge with the aid of the generalised Bernoulli Equation.

With known geometry, specified discharge and progress of the energy gradient, the respective water level position can be determined only by iteration (see Appendix A3), whereby in the case of structured cross-sections it is additionally complicated in that the relevant velocity head and the kinetic energy component can be given only with known discharge distribution.

Assuming a hydrostatic pressure distribution in all partial cross-sections (parallel flow) the energy head results from

∫=

α+=⋅⋅

+=n

1I

2

NN3iNNE g2

vhdAvQg2

1hh (29)

In this n is the number of partial cross-sections.

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As characteristic value for the degree of irregularity of the distribution of the velocity the correction value α in Eqn. 29 can be estimated in a simplified form from

∑ ⋅⋅

=α i3i2 Av

vQ1 (30)

With single cross-sections α = 1, with structured cross-sections α can lie between 1 and 2. With cross-sections normally used in wastewater engineering α = 1 may be set.

Taking different partial discharge cross-sections into account leads to the following discharge equation 11)

∑ ⋅= 2/1Ei JcQ (31)

with

c hydraulic control value of a partial cross-section i JE energy gradient

For the calculation of the flow process, the hydraulic control values are to be determined separately for the partial discharge cross-sections (see also Fig. 6). Depending on the selection of resistance law or flow formula you obtain the following conditional equations:

Fig. 6: Designations in the structured flow cross-sections

Calculation in accordance with Prandtl-Manning, Eqn. 9 - completely rough area

( )ii,hyi,hyii k/r84.14lg2r4g2Ac ⋅⋅⋅⋅⋅= (32)

Calculation in accordance with Manning-Strickler, Eqn. 18

3/2i,hyi,Stii rkAc ⋅⋅= (33)

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For three-part structured profile cross-sections (left - Index L; middle, channel - Index Ch; right - Index R) the following discharge equation is, for example, to be derived from Eqns. 31 and 33

( ) 2/1E

3/2R,hyR,StR

3/2Ch,hyRi,StCh

3/2L,hyL,StL JrkArkArkAQ ⋅⋅⋅+⋅⋅+⋅⋅= (34)

The iterative solution of this equation produces the discharge distribution in three-part structured flow cross-sections.

4 Dimensioning and Performance Verification

The calculations for dimensioning and for performance verification take place using the same hydraulic bases which, with regard to flow losses (roughness, individual losses) are described below in detail in 4.1 and 4.2. The procedure for the calculation range are described in 4.3 and 4.4.

4.1 Determination of Roughness

4.1.1 General Measure of Roughness k

Every calculation which makes use of the Prandtl-Colebrook resistance law requires knowledge of the equivalent sand roughness or the natural roughness of the inner wall of the channel in question. This degree of roughness is to be determined beforehand, as a rule on the basis of experimental data, if required, however, also through an appropriate estimation or determination on the basis of operational experience. The same applies for the roughness coefficients of each type of flow formula thus, for example, also for the coefficient kSt of the Manning-Strickler flow formula, whose arithmetic value corresponds with the degree of roughness k of the Prandtl-Colebrook resistance law (see Appendix A7). The special cases of deviating types of roughness (sand roughness, ripple roughness) are dealt with in Appendix A6.

If one solves the Prandtl-Colebrook resistance law for technical roughness behaviour (transitional region)

⋅+

λ⋅⋅−=

λ hyr84.14k

Re51.2lg21 (10)

for k, you obtain

λ⋅−⋅= λ−

Re51.210r84.14k )2/(1

hy (35)

as determining equation for the degree of roughness k

with ν

⋅= hyr4v

Re (36)

2Fhy

vJr4g2 ⋅⋅

=λ (37)

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⋅

−⋅−=b/Ag

v1dxdhJJ

2

SoF (38)

The relationship is to be evaluated using local measured data and, with k, provides a local, theoretically founded degree of roughness of the wetted channel sides. If the channel sides display technical roughness as assumed based on the Prandtl-Colebrook resistance law, then a sufficiently constant k value is to be expected independent of the experimental data used.

4.1.2 Operational Roughness kb

If local flow resistances or energy head losses together with losses as a result of wall friction are to be reflected in an augmented coefficient of roughness, together with losses resulting from wall friction, i.e. if discontinuous flow resistances or losses are to be included in an operational roughness kb, this can be done in the manner shown in Sect. 2.2, Eqn. (5a) ff. If the loss coefficients ζ of all disturbance sources located in the area of a channel reach l are known, and if λ is used to represent the resistance coefficient resulting from natural roughness and λb that resulting from operational roughness, the following definition equation applies:

∑ζ⋅+λ=λlr4 hy

b (6)

If the operational coefficient of resistance λb is calculated, knowing Re, k, rhy and Σζ, and if λ and Re are introduced into the resistance law, then kb/4rhy and from this the degree of roughness kb sought can be obtained. This increased degree of roughness produces mathematically the same total falls in energy as if one had applied continuous and local energy falls separately and then combined these. This operational degree of roughness kb is not dependent alone on the actual wall friction k and the incorporated individual resistances Σζ, but additionally on rhy,l as well as normally on the Reynolds Number Re. Global determination for kb must take this situation into account.

Under these preconditions it is possible and permitted, for dimensioning, for certain combinations of types of loss to function with a global value kb for operational roughness. Classification and justification are according to Table 4 dependent on the various types of sewer. Attention is drawn to Sect. 4.3 for preconditions and limitations.

Table 4: Global values for operational roughness kb [mm]

Type of sewer Design of shaft

Control shafts

Shaped shafts

Special shafts

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Transport sewer Main sewers ≤ DN 1000 Main sewers > DN 1000 Brickwork sewers, on-site concrete sewers, sewers made from non-standard pipes without special verification of roughness

0.50 0.75

- 1.50

0.50 0.75 0.75 1.50

0.75 1.50 1.50 1.50

Throttle lengths (1), pressure pipelines (1,2,3), siphons (1) and relining lengths without shafts

0.25

1) Excluding inlet, outlet and bend loses 2) Excluding pressure network (see also Sect. 9) 3) Effects on pumping stations see Sect. 9

4.2 Calculation of Individual Losses

The method presented in Sect. 4.1 for dealing with questions associated with the so-called operational roughness contains an detailed analysis of all individual influences from which those variables arise which can be combined mathematically into a kb value. With knowledge of the individual influences it is of course possible to collate deviating preconditions into an individual and justified treatment. For this purpose the following evaluation of available details and documents serves for individual losses as a result of

- positional inaccuracies and modifications, - pipe conditions, - inlet fittings, - shaft structures of standard design (straight passage)(12)

, - shaft structures of special design (straight passage), - curved structures and - conjunction structures.

The loss coefficients for this refer to

g2vh

2

i,f ⋅ζ= (4c)

__________________ 12) ATV Arbeitsblatt [Standard]-A 157

4.2.1 Loss Coefficients as a Result of Positional Inaccuracies and Changes (ζ’Pi)

The loss coefficients ζPi are to be taken from Table 5. They apply for the possible positional inaccuracies and changes for each pipe connection. With n points of connection the loss coefficient ζPi is to be applied n times.

Table 5: Loss coefficients ζPi

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DN ζPi

100 125 150 200 250 300 400 500

600 - 1000 > 1000

0.023 0.022 0.020 0.017 0.015 0.014 0.012 0.010 0.005

0

4.2.2 Loss Coefficients for Pipe Connections (ζPC)

The loss coefficients ζPC are to be taken from Table 6. They apply for the possible positional inaccuracies and changes for each pipe connection. With n points of connection the loss coefficient ζPC is to be applied n times.

Table 5: Loss coefficients ζPC

DN ζPC

100 125 150 200 250 300 400 500

600 - 1000 > 1000

0.020 0.016 0.012 0.009 0.007 0.006 0.004 0.003

0.0015 0.001

4.2.3 Loss Coefficients at Inlet Fixtures (ζin)

The loss coefficients ζin are to be taken from Table 7. They apply for every position at which an inlet fixture in the main sewer is planned or is present. With n such points of inlet domestic connections, road gullies) the loss coefficient ζin is to be applied n times. These loss coefficients cover only the effects of the geometry of the inlet fittings; the hydraulic effects of the inlet flow on the flow in the main sewer are covered in Sect. 5

Table 7: Loss coefficients ζin

HdZ ζin

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.000 0.002 0.004 0.007 0.011 0.016 0.022 0.029 0.036 0.045

2in

in Hd045.0

⋅=ζ (39)

with

din diameter of an inlet pipeline H profile height of main sewer

4.2.4 Loss Coefficients for Control Shafts (ζC)

Control shafts within the meaning of this Standard are shafts whose benching reaches up to the crown of the outgoing pipe (see Fig.7).

The values listed in Table 8 characterise the losses with subcritical discharge by ζ values gained experimentally. Slight impounding leads to considerable increase of the losses. All losses contain every influence as a result of losses with inflow, throughflow and outflow, however, not the pipe friction component over the length of the shaft. The loss for respectively one shaft is to be applied per sewer reach.

The ζ values for shafts with straight passage can be applied up to a change of flow direction of maximum 10° within a shaft using

ζC = 0.05 to 0.25

Shafts with change of direction of > 10° to 45° are to be taken into account using a ζ value of

ζC = 0.1 - 0.7.

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Fig. 7: Control shaft with raised benching

Table 8: Loss coefficients ζC

ζC for changes of direction of

Discharge situation d

h

0 - 10° >10° - 45° >45°

Partial filling < 10 0.05 0.1 0.15

Crown filling 1.0 0.1 0.2 0.5

Impounding, complete

filling under overpressure

> 1.0 0.25 0.7 1.0

Channel with cover plate

for all h/d 0.05 0.1 0.2

The values in Table 8 are applicable for the relative shaft diameter BSh/d and deflection radii r/d carried out with this normal in sewer systems. In the absence of detailed

ATV-DVWK-A 110E

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investigations the above given values can, until further notice, also be applied for shafts in sewers with non-circular profiles (oval profile, tapering profile).

4.2.5 Loss Coefficients for Special Shafts (ζS)

Special shafts within the meaning of this Standard are shafts in which the benching does not reach up to the crown of the outgoing pipe and with which, therefore, higher loss coefficients are to be applied mathematically.

Fig. 8: Special shafts with low-lying benching

The values listed in Table 9 characterise the losses with subcritical discharge by ζ values gained experimentally. Slight impounding leads to considerable increase of the losses. All losses contain every influence as a result of losses with inflow, throughflow and outflow, however, not the pipe friction component over the length of the shaft. The loss for respectively one shaft is to be applied per sewer reach.

The ζ values for shafts with straight passage can be applied up to a change of flow direction of maximum 10° within a shaft using

ζS = 0.15 to 0.85

Shafts with change of direction of > 10° to 45° are to be taken into account using a ζ value of

ζS = 0.3 - 1.3.

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In the absence of detailed investigations the above given values can, until further notice, also be applied for shafts in sewers with non-circular profiles (oval profile, tapering profile). The profile height H appears in the place of the diameter d.

Table 9: Loss coefficients ζS

ζS for changes of direction of

Discharge situation d

h

0 - 10° >10° - 45° >45°

Partial filling < 10 0.15 0.3 0.5

Crown filling 1.0 0.3 0.4 1.0

Impounding, complete

filling under overpressure

> 1.0 0.85 1.3 2.5

Channel with cover plate

for all h/d 0.05 0.1 0.2

The values in Table 8 are applicable for the relative shaft diameter BSh/d and deflection radii r/d carried out with this normal in sewer systems.

The values according to Table 8 for bends of 0 - 10° are to be applied for coupled shafts.

4.2.6 Loss Coefficients (ζfd) and Verifications for Flow Diversion

In the case of flow diversion, with regard to the verification to be carried out, one must differentiate strictly between subcritical discharge (Fr < 1) and supercritical discharge (Fr > 1).

For subcritical discharge loss coefficients ζfd can be determined and included in the hydraulic verification (see Sect. 4.2.6.1).

For supercritical discharge this procedure is not possible. The verification losses is replaced by design and operational details, which are to guarantee a disruption-free flow without unwanted effects on operation (see Sect. 4.2.6.2).

Transcritical discharge conditions with

0.75 < Fr < 1.5

are hydraulically unstable and therefore are, as far as possible, to be excluded (see also ATV Standard ATV-A 111, Sects. 3.2 and 3.6).

4.2.6.1 Subcritical Flow

The additional loss as a result of diversion with a bend radius of r referred to the sewer axis can be applied using

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ϕ⋅⋅=ζ sinrd

32 (40)

with

r average diversion radius ϕ diversion angle.

Loss coefficients as a result of flow diversion in shafts are already contained in the values in Tables 8 and 9.

4.2.6.2 Supercritical Flow

The flow in shafts with diversion with supercritical discharge is characterised less by resultant additional losses rather much more by wave formation at the outer edge of the bend, whereby the inner edge remains quasi “dry”. This influence rises with increasing Froude Number.

The wave height can be greater than the diameter of the exiting pipe. The wastewater then strikes against the shaft outlet wall and can lead to a collapse of the supercritical flow. Then a hydraulic jump occurs in the shaft which can move into the inlet pipe and which places the shaft under pressure discharge. The air inflow in the in the submerged pipe is then cut off; pulsation can then occur which effects a build up of the water level in the shaft and leads, finally, to the familiar phenomena of lifting of the manhole cover and increase of the noise loading.

The following conditions, which are fundamentally to be observed, result from detailed investigations into this13):

- the loading of the inlet sewer (QT/QV) may not exceed 50 % - the design of the shaft channel is always to be with raised benching (control shafts in

accordance with Sect. 4.2.4. - diversions of 90°, with the same r/d lead to smaller disturbances than with 45° (and

shorter layout of the flow path) - diversions of 45° can be held operationally more stable with an open straight

extension (l ≅ 3do)of the shaft - covers in the area of the shaft are of particular advantage, especially with

retrofitting/rehabilitation of existing facilities.

__________________ 13) ATV: Schießender Abfluß in Krümmerschachten. Abschlußbericht des Forschungsvorhanens (ATV 09/98) [Supercritical

discharge in elbow shafts. Final Report]

For the flow in shaft structures verification of

height (hM)

and

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position (θ)

of the wave maximum is to be carried out for Fro.

Fig. 9: Unobstructed elbow shaft; a) plan, b) longitudinal section

( )22o

o

M B50.01hh

⋅+= (41)

2oM )Fr(8.2tan ⋅ρ⋅=θ (42)

Here

2/1oo FrB ρ⋅= (43)

B - Bend Number

ρ = r/do (44)

ρ - relative bending of the shaft diversion referred to the diameter do of the inlet pipe and

r - radius of the elbow axis

For verification the following applies:

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uM dh <

and θ < ϕ

In addition the energy head in the shaft

(ho + vo2/2g)

is to be determined and compared with the upper surface of the ground.

With the non-observance of the above-given requirements either

- a covering using a cover plate in the shaft structure

or

- the modification of the shaft dimensions

has to be planned.

4.2.7 Loss Coefficients for Combining Structures (ζCS) and Verification for Flow Merging

The effects of losses in combining structures with subcritical flow are not included in the kb values according to Table 4. Verification in individual cases is always to be carried out(see also SIA 40).

4.2.7.1 Subcritical Discharge

For the loss coefficients

13CSζ loss coefficients for the flow from Sewer 1 to Sewer 3 in

)CS(f 13h energy head loss as a result of flow combining of Trains 1 and 3

)CS(f 13h = g2/v2

3LC13⋅ζ (45)

23CSh loss coefficients for the flow from Sewer 2 to Sewer 3 in

)CS(f 23h energy head loss as a result of flow combining of Trains 2 and 3

)CS(f 23h = g2/v2

3CS23⋅ζ (46)

applies with14)

X,Y,Z,a1,a2 auxiliary parameters ß1,ß2 combining angle Qi flow in Sewer i always larger than zero! Ai flow cross-section in Sewer i [m2]

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vi average velocity in Sewer i [m/s] i sewer indexing (i = 1;2;3) [1]

Table 10: Auxiliary parameters of loss coefficients for combining structures15)

ß1[°] ß2[°] 0 10 20 30 40 50 60 70 80 90

X(ß1) X(ß2) 0.95 0.95 0.95 0.94 0.94 0.93 0.90 0.82 0.73 0.63 a1(ß1) a2(ß2) 1.00 0.97 0.90 0.80 0.68 0.56 0.45 0.35 0.26 0.19

ß1[0] ß2[0] 100 110 120 130 140 150 160 170 180

X(ß1) X(ß2) 0.58 0.55 0.53 0.53 0.52 0.51 0.51 0.50 0.50 a1(ß1) a2(ß2) 0.15 0.12 0.11 0.12 0.16 0. 21 0.28 0.37 0.48

ZAQAQX1

2

13

31CS13

−

⋅⋅

⋅+=ζ (47)

ZAQAQY1

2

23

32CS23

−

⋅⋅

⋅+=ζ (48)

and the auxiliary quantities according to Table 10 and the auxiliary relationship for Z in accordance with Eqn. (49

⋅

⋅+⋅

⋅⋅=

2

32

3

22

1

32

3

11 A

AQQa

AA

QQa2Z (49)

In the absence of detailed investigations the above relations can be applied until further notice also for combining structures in sewers with non-circular profiles (oval cross-section, tapering cross-section).

___________________ 14) Note: this calculation instruction is based on the combined employment of the principle of linear momentum, equation of energy

and equation of continuity for “short structures” (equation of supporting force premises) 15) Prof. W thiedt, TH Damstadt

ATV-DVWK-A 110E

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Fig. 10: Designations in combining structures

4.2.7.2 Supercritical Discharge

No final details can be given in the relationships in combining structures with supercritical discharge as the ATV-DVWK research project on this is not yet completed. From previous available results16) findings are to be expected which require the treatment of this section similarly as under Sect. 4.2.6.2. In order not to delay the publication of this Standard, a detailed report will be published in the near future from which a formulation for this section can be presented to the specialist world for discussion. The amendment of this section will be implemented at the end of this discussion.

4.2.8 Other Loss Coefficients

Attention is drawn to Sect. 8 and the Literature for losses as a result of cases other than those dealt with.

4.3 Dimensioning

For the calculation case of dimensioning it is recommended that the global concept described in Sect. 4.1.2 is employed.

Here, it is emphasised that, with the global concept offered, the utilisation of the kb values in accordance with Table 4 for standard pipes without further verification is permitted and is to be taken as the standard case. Within the framework of the global approach with dimensioning, the effective wall roughness for pipes currently standardised by the DIN Standards Committee for Hydroscience is set uniformly with k = 0.1 and the flow rate as v = 0.8 m/s in order, with this also to cover the area of partial filling. For non-standard pipes without special verification, the effective wall roughness both for brickwork and local concrete sewers is to be set as kb = 1.5 mm.

_________________ 16) See trials report in KA 04/2001

The global approach for kb values as a rule includes the influences of

ATV-DVWK-A 110E

September 2001 39

- wall roughness, - positional inaccuracies and changes, - pipe connections, - inlet fittings and - shaft structures.

Values according to Table 4 include the influences of shaft structures up to and including crown filling h/d ≤ 1.0 and are therefore applicable without limitation with dimensioning.

Not included in this global definition of kb values are the influences of

a) undercutting of nominal widths, b) effects of impounding and overdamming, c) combining structures and d) inlet and outlet structures of throttle stretches, pressure pipelines and siphons.

In these cases the following principles apply:

To a)

- one should calculate using the effective mean clear width or the mean clear space; - with the dimensioning (planning), undercutting within the scope of DIN 4263 is

permitted and, within the sense of Sect. 3.1, is covered through the design with 0.9 Qv.

To c)

- the losses in combining structures are to be verified in the individual case; - verification can be dispensed with if a bottom drop ∆z of

20dz ≥∆

with d the diameter or profile height H of the outgoing sewer

is planned or exists; further a verification can be dispensed with if the loading of the continuing sewer, within the sense of Sect. 3.1, is limited with some 0.85 Qv instead of some 0.9 Qv, as long as this sewer is operated without back-up.

4.4 Performance Verification

For the performance verification of wastewater networks (existing or in planning) the

individual concept

is to be used, i.e. detailed taking into account of all loss influences in the individual case.

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Alternatively a suitably matched

synthetic individual concept

can also be used as the basis for the network to be calculated. Here kb values can be calculated and applied by reach or by area. An example calculation is contained in Appendix A8 for dealing with the individual case.

Depending on the impounding or overdamming which occurs, the loss coefficients for the shaft structure are to applied separately or by reach with the values for h/d > 1.0.

With the employment of the individual concept the losses as a result of wall roughness k and the individual losses arising are to be verified by reach, whereby k is fundamentally to be applied, also with regard to the change of characteristics of the pipe wall in long-term operation.

With performance verification of existing networks, if the effective clear width in the individual case is not or cannot be determined, one is to reckon basically with 95 % of the nominal width in which the cross-section reduction as a result of normal depositing is also covered.

With this, a special general kb value table for impounding, overdamming and flooding verification is impossible.

This procedure converts DIN EN 752-4, Sect. 9.2.3.

5 Flow with Lateral Inflow (Discontinuous Flow)

In sewer networks calculation is to be carried out along a calculation stretch, for example between two shafts with a flow increase as a result of lateral discharges (domestic connections, street gullies). An exception is formed only by pure transport sewers, throttle lengths and pressure pipelines. With collectors with lateral inflow, contrary to the details in Table 6, one has to work with a formulation for discontinuous flow. This requires the following system of equations:

Motion equation FSo JJdxdh

Agqvm −=+

⋅⋅

⋅ (50)

Continuity equation qdxdQ

= (51)

In the differential equation for the water surface profile to be derived from this, the lateral inflow then has the form that the right-hand side of this equation has to be expanded by an additional term to take into account change of the velocity head with which the energy which is required for the acceleration of the laterally discharged volume flow to the velocity of the main stream, is taken into account. Due to the lateral inflow the velocity distribution in the flow cross-section is also modified so that the previously employed formulation for the friction losses is no longer sufficient. The resistance behaviour, due to the complex flow, up to now could only be recorded in special cases. With the lack of sufficiently sound

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findings the additional loss can, in comparison with continuous flow, be taken into account approximately through the repeated approach of the modification of the velocity head. In general this results in

2qFSo

2

2FSo

Fr1JJJ

Fr1Ag

qQmJJ

dxdh −−

=−

⋅⋅⋅

−−= (52a)

with m as factor with the inclusion of additional losses (see Table 2). Due to the formulation described previously, m = 2 and thus

2qFSo

2FSo

2

22FSo

Fr1JJJ

Fr1Ag

qQ2JJ

Fr1AgqQ

AgqQJJ

dxdh −−

=⋅

⋅−−

=−

⋅⋅

−⋅⋅

−−= (52b)

The application of this formulation is not permitted for combining structures!

The following statements apply for the case of steady state, simplified non-uniform discontinuous flow. They can be transferred analogously to other types of calculation in accordance with Table 2.

5.1 Effect of Lateral Inflow

The flow loss depends on the following factors:

- total throughflow Q [m3/s], - size and distribution of the lateral inflow q [m3/(s.m)], - flow cross-section (with all partial influences), - bottom gradient and flow rate.

With unfavourable conditions - i.e. in particular if a large lateral inflow is discharged into a relatively small main stream over a short sewer length - the loss element Jq can reach several times JF.

5.2 Simplified Procedure

Depending on the tasking, simplified procedures are recommended for the application of Eqn. 52. These are different for the planning stage (dimensioning) and for the verification of existing networks (performance verification).

5.2.1 Dimensioning (Selection of a Constant Replacement Flow)

In order to avoid the costly evaluation of Eqn. 52 the energy head loss along a calculation stretch for a collector sewer, as a rule, is so determined that one determines the friction loss for a constant - that is not discontinuous - assumed flow (equivalent flow).

On the other hand, with the application of the throughflow Qe at the end of the calculation stretch as constant equivalent flow over the complete reach, a higher friction loss is

ATV-DVWK-A 110E

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determined with which the usually additional discharge loss of discontinuous flow is recorded.

Within the scope of justifiable accuracy, one can calculate using this simplification if, for the component ∆Q of the lateral inflow along a reach, the criteria in accordance with Table 11 are met in the various nominal width ranges.

Table 11: Limits on the validity of the calculation using Qe.

Relative lateral inflow

Nominal width range ∆Q/Qe

DN 200 to DN 500 DN 600 to DN 1000 DN 1100 to DN 2000

DN> 2000

No limitation ≤ 0.30 ≤ 0.10 ≤ 0.05

∆Q = Qe - Qi

With the exceeding of the validity limits for the equivalent flow the relationships presented using Eqn. 52 are to be investigated17) and, if required, a higher equivalent flow is to be employed.

5.2.2 Performance Verification

For the performance verification it can be assumed that the inventory of the network for which the verification is to be carried out has, in all respects, been recorded. This includes the position and admission of the lateral inflow from private properties and street drainage. With this it would be possible to record analytically the individual conditions existing at all discharge points and to deviate from the assumed evenly distributed lateral inflows according to Eqn. 52. This presumes that for each discharge point the geometric and hydraulic conditions are described and taken into account via an additional individual loss which has to be applied.

The combination of the described effects by reach and the application of an equivalent flow remains an option.

6 Flat Stretches and Depositing

Wastewater is a mixture of water with most non-uniform substances amongst which settleable matter is also to be met. Its sedimentation within the pipeline system can be prevented through suitable selection of the relevant parameters. A traverse wall stress of τ = 1.0 N/m2 should, under no circumstances, be undercut. Attention is drawn to the Literature for the fundamental detailss18).

__________________ 17) Ueker, K.J.: Abflußrechnung in Abwasserkanälen unter berücksichtigung seitlicher Einflüsse [Flow calculation in sewers taking

into account lateral inflows]. 18) Macke, E.: Über Feststofftransport bei niedrigen Konzentrationen in teilgefüllten Rohrleitungen [On solid matter transport with low

concentrations in partially filled pipelines].

Deposits are prevented if a necessary minimum wall traverse stress, which is dependent on the volume concentration of settleable solid matter, is achieved or exceeded. The

ATV-DVWK-A 110E

September 2001 43

necessary minimum wall traverse stress τmin in N/m2 , for concentrations of CT = 0.05 ‰ for combined wastewater and stormwater and CT = 0.03 ‰ for wastewater

τmin = 4.1Q1/3 (for stormwater and combined sewers) (53a)

τmin = 3.4Q1/3 (for normal sewers) (53b)

with Q in m3/s and that is independent of diameter and gradient of the pipeline considered.

The respectively available wall traverse stress τavail is calculated from

Fhyavail Jrg ⋅⋅⋅ρ=τ (54)

Assuming an operational roughness kb = 1.5 mm lower critical values Jc of the sole gradient for the different nominal width ranges of oval profiles and degrees of filling of hT/d = 0.1 to 0.5 and for τ ≥ 1.0 N/m2 according to Tables 12a and 12b19).

Both tables also contain ranges which are characterised by the maintenance of τmin = 1.0 N/m2. The details for these are underlaid in grey. For filling heights of h < 3 cm the conditions of an even concentration with steady state discharge no longer exist. In these cases it is recommended that gradients are determined with J ≥ 1:DN with DN in mm.

The critical values of Tables 12a and 12b can be applied with sufficient accuracy for all kb values = 0.25 mm to kb = 1.5 mm18).

If, in the individual case, other values for cT are relevant then attention is drawn to evaluation in accordance with footnote 18), whereby higher concentrations lead to greater critical velocities. These are not to be seen as permitted minimum velocities but rather as indication of the occurrence or the prevention of deposits.

With the as a rule non-constant discharges, the discharge Q relevant for this consideration is determined in that the time with deposits (τavail < τmin) is no longer double the time without deposits (τavail ≥ τmin).

The values of Tables 12a and 12b can also be used for profiles which, in the area of the sole, do not significantly deviate from circular profiles. Here d = B/2 applies for oval profiles and d = B for tapering cross-sections.

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Table 12a: Limiting values for deposit-free operation of stormwater and combined wastewater sewers

Circular x-section d

hT/d ≥ 0.10 hT/d ≥ 0.20 hT/d ≥ 0.30 hT/d ≥ 0.50

Jc vc τmin Jc vc τmin Jc vc τmin Jc vc τmin mm ‰ m/s n/m2 ‰ m/s n/m2 ‰ m/s n/m2 ‰ m/s n/m2

200 *) *) *) 4.23 0.43 1.00 2.98 0.46 1.00 2.04 0.48 1.00 250 *) *) *) 3.38 0.45 1.00 2.39 0.47 1.00 1.63 0.49 1.00 300 5.35 0.43 1.00 2.82 0.46 1.00 1.99 0.49 1.00 1.48 0.53 1.09 350 4.59 0.44 1.00 2.42 0.47 1.00 1.70 0.50 1.00 1.45 0.58 1.24 400 4.02 0.44 1.00 2.11 0.48 1.00 1.61 0.51 1.05 1.42 0.63 1.39 450 3.57 0.45 1.00 1.88 0.49 1.00 1.53 0.55 1.15 1.40 0.67 1.54 500 3.21 0.46 1.00 1.69 0.50 1.00 1.50 0.59 1.26 1.38 0.71 1.69 600 2.68 0.47 1.00 1.61 0.54 1.14 1.47 0.66 1.48 1.34 0.79 1.97 700 2.29 0.48 1.00 1.59 0.61 1.32 1.43 0.71 1.68 1.31 0.86 2.25 800 2.01 0.49 1.00 1.55 0.64 1.47 1.40 0.77 1.88 1.29 0.93 2.52 900 1.88 0.51 1.05 1.52 0.68 1.62 1.38 0.82 2.08 1.26 0.99 2.79 1000 1.84 0.54 1.15 1.50 0.73 1.78 1.36 0.87 2.28 1.24 1.05 3.05 1100 1.81 0.56 1.24 1.48 0.77 1.93 1.35 0.93 2.49 1.23 1.11 3.31 1200 1.79 0.60 1.34 1.46 0.81 2.07 1.32 0.96 2.66 1.21 1.17 3.57 1300 1.77 0.63 1.43 1.44 0.84 2.22 1.30 1.00 2.84 1.20 1.22 3.82 1400 1.75 0.65 1.53 1.43 0.88 2.37 1.30 1.06 3.05 1.18 1.27 4.07 1500 1.73 0.67 1.62 1.41 0.91 2.50 1.28 1.09 3.22 1.17 1.32 4.31 1600 1.71 0.71 1.70 1.40 0.95 2.65 1.27 1.12 3.39 1.16 1.37 4.55 1800 1.69 0.75 1.89 1.38 1.01 2.93 1.25 1.22 3.77 1.14 1.46 5.03 2000 1.66 0.79 2.06 1.36 1.07 3.22 1.23 1.28 4.11 1.12 1.54 5.50 2200 1.64 0.83 2.24 1.34 1.13 3.48 1.21 1.35 4.46 1.11 1.63 5.97 2400 1.61 0.86 2.41 1.32 1.18 3.74 1.19 1.41 4.80 1.09 1.70 6.42 2600 1.59 0.92 2.58 1.30 1.23 3.99 1.17 1.45 5.11 1.08 1.78 6.87 2800 1.58 0.96 2.75 1.29 1.29 4.27 1.16 1.52 5.45 1.07 1.85 7.32 3000 1.56 0.99 2.92 1.27 1.32 4.50 1.15 1.58 5.78 1.05 1.92 7.76 3200 1.54 1.01 3.07 1.26 1.37 4.78 1.14 1.64 6.11 1.04 1.99 8.19 3400 1.53 1.05 3.24 1.25 1.42 5.01 1.13 1.70 6.44 1.03 2.05 8.62 3600 1.51 1.07 3.39 1.24 1.46 5.27 1.12 1.74 6.74 1.03 2.12 9.05 3800 1.50 1.11 3.56 1.22 1.49 5.48 1.11 1.82 7.09 1.02 2.18 9.47 4000 1.49 1.16 3.73 1.21 1.54 5.75 1.10 1.85 7.39 1.01 2.24 9.89 *) J ≥ 1/DN

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Table 12b: Limiting values for deposit-free operation of normal sewers Circular

x-section d hT/d ≥ 0.10 hT/d ≥ 0.20 hT/d ≥ 0.30 hT/d ≥ 0.50

Jc vc τmin Jc vc τmin Jc vc τmin Jc vc τmin mm ‰ m/s n/m2 ‰ m/s n/m2 ‰ m/s n/m2 ‰ m/s n/m2

150 *) *) *) 5.64 0.41 1.00 3.98 0.44 1.00 2.72 0.45 1.00 200 *) *) *) 4.23 0.43 1.00 2.98 0.46 1.00 2.04 0.48 1.00 250 *) *) *) 3.38 0.45 1.00 2.39 0.47 1.00 1.63 0.49 1.00 300 5.35 0.43 1.00 2.82 0.46 1.00 1.99 0.49 1.00 1.36 0.51 1.00 350 4.59 0.44 1.00 2.42 0.47 1.00 1.70 0.50 1.00 1.18 0.52 1.01 400 4.02 0.44 1.00 2.11 0.48 1.00 1.49 0.51 1.00 1.16 0.56 1.13 450 3.57 0.45 1.00 1.88 0.49 1.00 1.33 0.52 1.00 1.14 0.60 1.26 500 3.21 0.46 1.00 1.69 0.50 1.00 1.22 0.53 1.03 1.12 0.64 1.37 600 2.68 0.47 1.00 1.41 0.51 1.00 1.20 0.59 1.20 1.09 0.71 1.61 700 2.29 0.48 1.00 1.30 0.55 1.07 1.16 0.63 1.36 1.07 0.78 1.83 800 2.01 0.49 1.00 1.26 0.58 1.20 1.14 0.69 1.53 1.05 0.84 2.06 900 1.78 0.50 1.00 1.25 0.63 1.33 1.12 0.73 1.69 1.03 0.90 2.27 1000 1.61 0.50 1.00 1.23 0.67 1.45 1.11 0.78 1.86 1.01 0.95 2.49 1100 1.49 0.52 1.02 1.21 0.69 1.57 1.09 0.82 2.01 1.00 1.00 2.70 1200 1.46 0.54 1.09 1.19 0.73 1.69 1.08 0.87 2.17 0.99 1.05 2.91 1300 1.45 0.56 1.17 1.18 0.77 1.82 1.07 0.92 2.33 0.98 1.10 3.11 1400 1.44 0.60 1.25 1.16 0.79 1.93 1.06 0.95 2.48 0.96 1.15 3.31 1500 1.41 0.61 1.32 1.16 0.83 2.05 1.04 0.98 2.62 0.96 1.19 3.51 1600 1.40 0.63 1.40 1.14 0.86 2.16 1.03 1.01 2.76 0.95 1.23 3.71 1800 1.38 0.68 1.55 1.12 0.91 2.38 1.01 1.07 3.05 0.93 1.31 4.10 2000 1.35 0.71 1.68 1.10 0.96 2.60 1.00 1.15 3.35 0.91 1.39 4.49 2200 1.34 0.76 1.83 1.08 1.01 2.82 0.99 1.22 3.64 0.90 1.47 4.86 2400 1.32 0.79 1.97 1.07 1.06 3.04 0.97 1.26 3.90 0.89 1.54 5.23 2600 1.30 0.82 2.10 1.06 1.11 3.25 0.96 1.33 4.18 0.88 1.61 5.60 2800 1.29 0.86 2.25 1.05 1.16 3.47 0.95 1.39 4.46 0.87 1.67 5.96 3000 1.27 0.88 2.37 1.04 1.20 3.67 0.94 1.43 4.72 0.86 1.73 6.32 3200 1.25 0.90 2.50 1.03 1.25 3.89 0.93 1.49 5.00 0.85 1.80 6.68 3400 1.24 0.94 2.63 1.02 1.29 4.10 0.92 1.53 5.25 0.84 1.85 7.03 3600 1.23 0.97 2.76 1.01 1.32 4.29 0.91 1.56 5.49 0.84 1.91 7.38 3800 1.23 1.01 2.91 1.00 1.36 4.48 0.90 1.62 5.76 0.83 1.97 7.72 4000 1.22 1.03 3.03 1.00 1.42 4.71 0.90 1.68 6.03 0.82 2.02 8.06 *) J ≥ 1/DN

7 Steep Stretches and Air Transfer Within the meaning of this standard steep stretches are understood to be sewer sections within which, with supercritical discharge, the air mixture is to be taken into account with dimensioning. This applies if the Boussinesq Number (Eqn. 56) is greater than 6.

With the hydraulic dimensioning of a steep stretch there are three different sections to be differentiated (see Fig.11):

- inflow with acceleration stretch without air uptake, - throughflow with air uptake and equilibrium stretch, - outflow with energy conversion.

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Fig. 11: Section of a steep stretch

7.1 Steep Stretches - Inflow

As a rule, the critical depth hcrit is passed through in the area of the inflow. Using a determination of the line of the water surface profile, an estimation is to be made as to how far the acceleration stretch reaches, i.e. in which area of the sewer the transition to the normal water depth with supercritical discharge is achieved.

Retention of the sewer cross-section above the steep stretch up to the end of the acceleration stretch or, if possible, creation of a continuous transition over the cross-section in the area of the throughflow of the steep stretch is recommended. Aim of this measure is the avoidance of pulsing discharge conditions in sewer operation.

7.2 Steep Stretches - Throughflow

In this area of a steep stretch, a build-up of a water-air mixture takes place with discharge (absorption stretch), whose safe discharge is to be guaranteed through the appropriate dimensioning of the sewer cross-section (equilibrium stretch).

For sole gradients greater than 200 ‰, the friction gradient is to be determined with the aid of actual length l of the sewer9). The values Q and v are then to be determined for a planned cross-section with complete filling as well as with partial filling. The admixture of air is determined using the enlargement ratio fair (according to Volkart, see Appx. A4). This is dependent on the flow rate, the flow cross-section and the operational roughness. The value relevant for dimensioning is

airTDIM fQQ ⋅= Q (55)

for which QDIM < 0.75Qv (corresponds with h/d < 0.65) is to be maintained.

With the aid of the Boussinesq Number Bou

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hy

T

rgvBou⋅

= (56)

the air concentration C is determined as

1)0.6Bou(02.011C 5.1 +−

−= (57)

The velocity vM of the water-air mixture is

)C1(vv 2TM −= (58)

The air component QAir is

( )C1/CQQ TAir −⋅= (59)

The filling height h of the water-air mixture in the area of the equilibrium stretch is calculated as

G

ATM V

QQA += (60)

and the relationship h/d or h/H, which can be taken from Appx. A2, Tables 14 to 19 for AM/AV (in place of AT/AV).

With consecutive sewer sections with changing gradient within a partial stretch, the air component of the above lying sections is maintained for the following sections.

The hydrodynamic forces which occur are to be calculated with change of direction within the throughflow stretches.

7.3 Steep Stretches - Outflow

The transition from supercritical to subcritical flow, as a rule, takes place in this area. The energy conversion with this is to be localised; forces occurring are to be absorbed. For this there are available various forms of energy conversion structures20) developed specially for this area.

With transition with hydraulic jump in the subsequent sewer section without closure the calculation of circular profiles takes place according to Eqns. 61 to 63.

41

T1

hdg

QFR⋅⋅

= (61)

The calculation of the Froude Number for the hydraulic jump in the subsequent sewer section takes place with the water depth h1 from hT (without taking account of the air transfer) assuming unchanged flow velocity in the throughflow and outflow cross-sections.

_______________ 20) ATV: ATV-Handbuch, Planung der Kanalisation [ATV Manual, Planning of the sewer system]

The ratio of the conjugated water depths before and after the hydraulic jump (Fig. 12) is:

ATV-DVWK-A 110E

September 2001 48

9.01

1

2 Frhh

= (62)

The length of the hydraulic jump is:

2/11

2

H Fr4hL

≅ (63)

The position of the hydraulic jump (Fig. 12) is determined through the calculation of a backwater curve. Effects of a backing-up of tail water are to be taken into account.

Fig. 12: Hydraulic jump in the steep section outflow (a: partial filling; b: complete filling)

With closure with h2 ≥ du the following applies

ATV-DVWK-A 110E

September 2001 49

41u1

2

hdg

Qhh

⋅⋅= (64)

The minimum diameter without closure is

3 2h

2

u hgQd⋅

≥ (65)

With hydraulic jump in the outflow area provision has always to be made for aeration and ventilation.

8 Special Structures

The handling of the hydraulic calculation for the special structures employed in the sewer system would exceed the scope of this set of rules and standards. Therefore there are individual standards created for these which are

ATV-A 111 for relief facilities - with the following detailing Facilities/structures for the influencing of discharge under open channel operation (overflows) Vertical approach flow Lateral approach flow (side weir) Approach flow in bends and from overfalls Siphon weirs Movable obstructors Leaping weir (stormwater overflow with bottom opening) Special designs

Facilities for the influencing of discharge under pressure Throttle devices Throttle lengths Throttle slide valves Eddy devices (e.g. throttles, valves) Outlet slots Regulating devices Control devices

and in ATV-A 112 all further structures not covered in ATV-A 110 or ATV-A 111 such as Siphons Air cushion siphons Drop structures

Drop structures with bypass Drop structures with chute Drop shaft with deflector or baffle

Vortex drop shafts Discharge structures

ATV-DVWK-A 110E

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Curved structures Diameter change in shafts

As supplement attention is drawn to SIA 40 - Documentation 40 - of the Swiss Engineers and Architects Association.

Attention is drawn to the literature for structures and constructions which go beyond the scope of this set of rules and standards.

9 Pressure and Vacuum Drainage, Compressed Air Flushed Wastewater Transport Pipelines, Wastewater Pumping Stations with Pressure Pipelines

On this subject the following European Standards are valid in the field of drainage systems outside buildings:

DIN EN 1671 Pressure sewerage systems outside buildings

DIN EN 1091 Vacuum sewerage systems outside buildings

These are to be supplemented in the future by residual standard specifications in the form of revisions of ATV Standard ATV-A 116 and are:

ATV-DVWK-A 116 Part 1 Vacuum drainage systems outside buildings

ATV-DVWK-A 116 Part 2 Pressure drainage systems outside buildings

The following will be dealt with in a further document:

ATV-DVWK-A 116 Part 3 Compressed air flushed wastewater transport pipelines

Up until now a report has been available on the latter subject, which was published in KA 1/87.

Within buildings the following are to be noted

DIN EN 12109 Vacuum drainage systems inside buildings

For wastewater pumping stations with pressure pipelines attention is drawn to

DIN EN 752 Part 6 Pumping installations

together with the Standard

ATV-DVWK-A 134 Planning and construction of wastewater pumping facilities

10 Private Property Drainage

The harmonisation, initiated by the EG (EU) Commission 1991, of sets of technical rules and standards in Europe (EU with EFTA, CEFTA and other associated members of CEN) for wastewater, differentiates

Drainage systems

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September 2001 51

- outside buildings, and - inside buildings.

For drainage systems outside buildings the follow series of standard specifications applies

DIN EN Parts 1 to 7,

within buildings

DIN EN 12056 Parts 1 to 6 apply.

There is a duty to take over the European standard specification so that

DIN 1986

previously alone applicable in Germany is to be adapted and reissued as a so-called residual standard specification.

In private property drainage systems there are connection-, drop-, collector- and house drains to be dimensioned. With this it is not only the undisrupted removal of discharged wastewater which has to be guaranteed, but also that the air carried can also be transported.

Table 13a: Permitted discharge, degree of filling 50 % (h/d = 0.5), wastewater21) Gradient DN 100 DN 125 DN 150 DN 200 DN 225 DN 250 DN 300 J Qmax v Qmax v Qmax v Qmax v Qmax v Qmax v Qmax v [cm/m] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] 0.5 1.8 0.5 2.8 0.5 5.4 0.6 10.0 0.8 15.9 0.8 18.9 0.9 34.1 1.0 1.0 2.5 0.7 4.1 0.8 7.7 0.9 14.2 1.1 22.5 1.2 26.9 1.2 48.3 1.4 1.5 3.1 0.8 5.0 1.0 9.4 1.1 17.4 1.3 27.6 1.5 32.9 1.5 59.2 1.8 2.0 3.5 1.0 5.7 1.1 10.9 1.3 20.1 1.5 31.9 1.7 38.1 1.8 68.4 2.0 2.5 4.0 1.1 6.4 1.2 12.2 1.5 22.5 1.7 35.7 1.9 42.6 2.0 76.6 2.3 3.0 4.4 1.2 7.1 1.4 13.3 1.6 24.7 1.9 39.2 2.1 46.7 2.2 83.9 2.5 3.5 4.7 1.3 7.6 1.5 14.4 1.7 26.6 2.0 42.3 2.2 50.4 2.3 90.7 2.7 4.0 5.0 1.4 8.23 1.6 15.4 1.8 28.5 2.1 45.2 2.4 53.9 2.5 96.9 2.9 4.5 5.3 1.5 8.7 1.7 16.3 2.0 30.2 2.3 48.0 2.5 57.2 2.7 102.6 3.1 5.0 5.6 1.6 9.1 1.8 17.2 2.1 31.9 2.4 50.6 2.7 60.3 2.8 108.4 3.2

_______________ 21) Source: DIN EN 12056-2:Gravity drainage systems within buildings, Part 2: Wastewater systems, planning and calculation

The dimensioning of collectors (within buildings) and house drains (outside and below buildings) takes place in accordance with DIN EN 12056 Part2, according to Prandtl-Colebrook for the respective relevant gradient; for this Tables 13a and 13b are provided.

Table 13b: Permitted discharge, degree of filling 70 % (h/d = 0.7), wastewater21) Gradient DN 100 DN 125 DN 150 DN 200 DN 225 DN 250 DN 300 J Qmax v Qmax v Qmax v Qmax v Qmax v Qmax v Qmax v

ATV-DVWK-A 110E

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[cm/m] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] [l/s] [m/s] 0.5 2.9 0.5 4.8 0.6 9.0 0.7 16.7 0.8 26.5 0.9 31.6 1.0 56.8 1.1 1.0 4.2 0.8 6.8 0.9 12.8 1.0 23.7 1.2 37.6 1.3 44.9 1.4 80.6 1.6 1.5 5.1 1.0 8.3 1.1 15.7 1.3 29.1 1.5 46.2 1.6 55.0 1.7 98.8 2.0 2.0 5.9 1.1 9.6 1.2 18.2 1.5 33.6 1.7 53.3 1.9 63.6 2.0 114.2 2.3 2.5 6.7 1.2 10.8 1.4 20.3 1.6 37.6 1.9 59.7 2.1 71.1 2.2 127.7 2.6 3.0 7.3 1.3 11.8 1.5 22.3 1.8 41.2 2.1 65.4 2.3 77.9 2.4 140.0 2.8 3.5 7.9 1.5 12.8 1.6 24.1 1.9 44.5 2.2 70.6 2.5 84.2 2.6 151.2 3.0 4.0 8.4 1.6 13.7 1.8 25.6 2.1 47.6 2.4 75.5 2.7 90.0 2.8 161.7 3.2 4.5 8.9 1.7 14.5 1.9 27.3 2.2 50.5 2.5 80.1 2.8 95.5 3.0 171.5 3.4 5.0 9.4 1.7 15.3 2.0 28.8 2.3 53.3 2.7 84.5 3.0 100.7 3.1 180.8 3.6

Attention is drawn to

DN EN 12056 Part 3

for roof drainage.

11 Cost Aspects

The consequent application of the dimensioning principles laid down in the standard aid the production of operationally suitable systems and, with this, the minimisation of subsequent costs. All calculation methods given contain sufficient information on justified deviations (opening stipulations). This applies in particular for the proposal of working using an individual or global concept with dimensioning or performance verification.

ATV-DVWK-A 110E

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12 Literature [Translator's note: known translations are give in English, otherwise a courtesy translation is provided in square brackets]

ATV ATV-Handbuch, Planung der Kanalisation [ATV Manual, Planning of the sewer system], 4th Edition, Verlag Ernst Sohn, 1994.

ATV Standard ATV-A 111E, Standards for the hydraulic dimensioning and performance verification of sewers and drains, Hennef, February 1994.

ATV Arbeitsblatt ATV-A 112, Richtlinien für die Hydraulische Dimensionierung und den Leistungsnachweis von Sonderbauwerken in Abwasserkanälen und -leitungen [Standard ATV-A 112, Standards for the hydraulic dimensioning and performance verification of special structures in sewers and drains], Hennef, January 1998.

ATV Standard ATV-A 116E, Special Sewer Systems, Sankt Augustin, September 1992.

ATV Standard ATV-A 118E, hydraulic dimensioning and verification of drainage , Hennef, November 1999.

ATV-DVWK Arbeitsblatt ATV-DVWK-A 134, Planung und Bau von Abwasserpumpanlagen [Standard ATV-DVWK-A 134, Planning and construction of wastewater pumping stations], Hennef, June 2000.

ATV-DVWK Arbeitsblatt ATV-DVWK-A 157, Bauwerke in Entwässerungsanlagen [Standard ATV-DVWK-A 157, Structures in drainage systems], Hennef, December 2000.

ATV Merkblatt ATV-M 165 Anforderungen an Niederschlags-Abfluß-Berechnungen in der Stadtentwässerung [Advisory Leaflet ATV-M 165, Requirements on precipitation discharge calculations in municipal drainage], Hennef, April 1994.

ATV Beeinflussung der Leistungsfähigkeit von Kanalstrecken durch konstruktiven Veränderungen im Bereich der Schächte [Influencing of the efficiency of sewer lengths through design changes in the area of shafts]. Final report of the research project (ATV 01/97), October 1999.

ATV-DVWK Schießender Abfluß im 45°-Vereinigungsschacht. Abschlußbericht des ATV-DVWK-Forschungsvorhabens [Supercritical discharge in a 45° combining shaft. Final report of the ATV-DVWK research project] (ATV-DVWK 01/2000).

ATV Druckluftgespülte Abwassertransportleitungen, Planungs-, Bau- und Betriebsgrundsätze; Arbeitsbericht der ATV-AG 1.1.6, [Compressed air flushed wastewater transport drains, Planning, construction and operational principles, report of the ATV WG 1.1.6] Korrespondenz Abwasser 1/87, p. 70 ff.

ATV Schießender Abfluß in Krümmerschachten. Abschlußbericht des Vorschungsvorhabens [Supercritical discharge in elbow shafts. Final report of the research project] (ATV 09/98), Korrespondenz Abwasser 8/99, p. 1247 ff.

DIN DIN EN 752, Drain and sewer systems outside buildings Part 1, Generalities and definitions Part 2, Performance requirements Part 3, Planning Part 4, Hydraulic design and environmental considerations Part 5, Rehabilitation Part 6, Pumping installations Part 7, Maintenance and operations.

DIN DIN EN 1071, Unterdruckentwässerungssysteme außerhalb von Gebäuden [Vacuum drainage systems outside buildings].

ATV-DVWK-A 110E

September 2001 54

DIN DIN EN 1671, Pressure sewerage systems outside buildings. DIN DIN 1986, Drainage systems on private ground

Part 2, Design of sizes of drain, waste and ventilation pipes. DIN DIN 4044, Hydromechanik im Wasserbau; Begriffe

[Hydromechanics in hydraulic engineering; Terms]. DIN DIN 4045, Wastewater engineering; Terms. DIN DIN 4263, Shapes, dimensions and geometrical values of sewers and drains for water

and wastewater engineering. DIN DIN EN 12056, Gravity drainage inside buildings

Part 1, General and performance requirements Part 2, Sanitary pipework, layout and calculation Part 3, Roof drainage, layout and calculation Part 4, Wastewater lifting plants. Layout and calculation Part 5, Installation and testing, instructions for operation, maintenance and use Part 6, (DRAFT - prEN) Inspection and testing.

DIN DIN EN 12109, Vacuum drainage systems inside buildings. Franke, P.-G.: Die Rauhigkeitsverhältnisse im teilgefüllten Rohr

[Roughness conditions in the partially filled pipe]. Die Wasserwirtschaft 46 (1956) No. 12, p. 315-318.

Schröder, R.C.M.: Hydraulische Methoden zur Erfassung von Rauheiten [Hydraulic methods of recording roughness] DVWK Publications No. 92, Paul-Parey-Verlag Hamburg und Berlin, 1990.

Gothe, E.; Valentin, F.:

Schachtverlust bei Überstau. [Shaft loss with additional impoundment] Korrespondenz Abwasser 4/92, p. 470 ff.

Hager, W.H.: Fließformeln für turbulente Strömungen [Flow formulas for turbulent flows], Wasserwirtschaft (1987), No. 12.

Hager, W.H.: Abwasserhydraulik. [Wastewater hydraulics] Springerverlag Berlin 1995.

Hager, W.H.; Del Guidice, G.; Gisonni, C.:

Schießender Abfluß in Krümmerschächten. Gefördert mit Mitteln des Forschungsfonds von ATV und GFA: [Supercritical discharge in elbow shafts. Sponsored with resources from the ATV and GFA research fund], KA Korrespondenz Abwasser, 8/99, p. 1267 ff.

Hager, W.H.; Gisonni, C.:

Schießender Abfluß im 45°-Krümmersschacht [Supercritical flow in the 45° bend shaft] KA Wasserwirtschaft, Abwasser, Abfall, 7/2000, p. 1047-1052.

Hager, W.H.; Del Guidice, G.:

Schießender Abfluß im 45°-Vereinigungsschacht [Supercritical flow in the 45° combining shaft] KA 4/2001.

Höfer, U.: Beginn der Sedimentbewegung bei Gewässersohlen mit Riffeln oder Dünen [Start of sedimentation movement with the bottom of surface waters with layers of shale or dunes] Technical report of the (German) Institute for Hydraulic Engineering (IHH), Technical University Darmstadt, No. 32, 1984.

Kölling, C.; Valentin, F.:

SIMK-Simulation von Teilgefüllungskurven. Gefördert von ATV und GFA und fachlich unterstützt durch die ATV-Arbeitsgruppe 1.2.2. „Hydraulische Berechnung von Kanälen und Leitungen“ [SIMK simulation of partial filling curves. Sponsored by ATV and GFA and technically supported by the ATV WG 1.2.2. “Hydraulic calculation of sewers and drains”]. KA Korrespondenz Abwasser, 3/99, p. 410 ff.

Kölling, C.; Valentin, F.:

SIMK-Simulation von Teilgefüllungskurven. Abschlußbericht des Forschungsvorhabens (ATV 25/97 and ATV 31/99) [SIMK simulation of partial filling curves. Final report of the research project (ATV 25/97 and ATV 31/99)]

ATV-DVWK-A 110E

September 2001 55

KA Korrespondenz Abwasser, 3/99, p. 410 ff. Macke, E.: Über Feststofftransport bei niedrigen Konzentrationen in teilgefüllten Rohrleitungen.

[On the transport of solid matter with low concentrations in partially filled pipelines] Information of the Leichtweiß Institute for Hydraulic Engineering Technical University Braunschweig, No. 69, Braunschweig 1980.

Merlein, J.; Valentin, F.:

Beeinflussung der Leistungsfähigkeit von Kanalstrecken durch konstruktive Veränderungen im Bereich der Schächte [Influencing of the efficiency of sewer lengths through design changes in the are of the shafts] KA, Wasserwirtschaft, Abwasser, Abfall, 8/2000, p. 1176-1181.

Naudascher, E.: Hydraulik der Gerinne und Gerinnebauwerke [Hydraulics of channels and channel structures] Springer-Verlag, Wien 1992.

Pecher, R.; Schmidt, H.; Pecher, D.:

Hydraulik der Abwasserkanäle in der Praxis [Hydraulics of sewers in practice] Parey-Verlag, Hamburg and Berlin 1991.

Sauerbrey, M.: Abfluß in Entwässerungsleitungen unter besonderer Berücksichtigung der Fließvorgänge in teilgefüllten Rohren. [Discharge in drainage pipelines take particular account of flow procedures in partially filled pipes]. Wasser und Abwasser in Forschung und Praxis, Vol. 1. Erich-Schmidt-Verlag, Bielefeld 1969.

Schröder, R. und Knauf, D.:

Über das hydraulische Widerstandsverhalten von Beton- und Stahlbetonrohren im Übergangsbereich [On the resistance behaviour of concrete and reinforced concrete pipes in the transition zone]. Das Gas- und Wasserfach 113, (1972), No.11, p.536-541.

SIA Sonderbauwerke der Kanalisationstechnik, Hydraulische Berechnungsgrundlagen und konstruktiven Hinweise [Special structures in sewer engineering, fundamentals of hydraulic calculation and design information] SIA documentation No. 40, Swiss Engineer and Architect Association, Zürich, 1980.

Tiedt, W.: Hydrodynamische Untersuchungen des Teilfüllungsproblems. [Hydrodynamic investigations of partial filling problems] Technical report of the German Institute for Hydraulics and Hydrology, Technical University Darmstadt No. 7, Darmstadt 1971.

Ueker, K.J.: Abflußrechnung in Abwasserkanälen unter Berücksichtigung seitlicher Zuflüsse [Discharge calculations in sewers taking into account lateral inflows]. Steinzeug (vitrified clay) Information 87, Specialist Association of the German Vitrified Clay Industry, Max-Planck-Str. 6, 50858 Köln (Marsdorf), 1987.

Unger, P.: Grundlagen der kb-Wert-Festlegung im Arbeitsblatt A 110 [Basics of kb value determination in ATV Standard ATV-A 110] Korrespondenz Abwasser 1/1989, p. 46 ff.

Volkart, P.: Hydraulische Bemessung steiler Kanalisationsleitungen unter Berücksichtugung der Luftaufnahme. [Hydraulic dimensioning of steep sewer pipelines taking into account air uptake]. Dissertation ETH Zürich, No. 6104, Zürich 1978, also appeared as Information of the Research Centre for Hydraulic Engineering, Hydrology and Glaciology, ETH Zürich, No. 30, Zürich 1978.

Volkart, P.: Hydraulische Bemessung teilgefüllter Steilleitungen [Hydraulic dimensioning of partially filled steep pipelines] Gas-Wasser-Abwasser 58 (1978), No. 11, p. 658-667.

Zanker, G.: Neues Schachtsystem nach DIN 4034 [New shaft system in accordance with DIN 4034] Abwassertechnik 3/87, p. 22 ff.

ATV-DVWK-A 110E

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13 Symbols, Units and Terms [Translator’s note: For ease of comparison symbols used in the German original, if different from those used in this English version, are in brackets after the English symbols]

Symbols α [1] Correction factor for non-uniform velocity distribution A [m²] Flow cross-section perpendicular to the sole AE(AG) [m²] Flow cross-section in the area of the equilibrium stretch

with the discharge of water-air mixture ß1,ß2 [°] Combining angle b [m] Water level breadth B [1] Bend number B [m] Profile breadth Bsh [m] Shaft breadth, shaft diameter Bou [1] Boussinesq Number ci [m3/s] Hydraulic conductance for partial cross -sections cS [1] Loss factor for lateral inflow cT [1] Solid matter concentration in wastewater C [1] Air concentration d [m] Circular pipe diameter (clear diameter CD) dhy [m] Hydraulic diameter din(dz) [m] Diameter of an inflow pipeline DN [mm] Nominal diameter, nominal width η [kg/m⋅s] Dynamic viscosity ϕ [1] Ratio for partial cross-sections i.a.w. DIN 4263 ϕ [°] Slope angle ϕ [°] Deflection angle f [1] Form coefficient fair(fluft) [1] Enlargement factor with discharge of water-air mixture Fr [1] Froude Number χ [kN/m3] Specific gravity of flow medium g [m/s2] Acceleration due to gravity h [m] Filling height, flow depth, pressure head at the pipe or

profile sole with completely filled flowing pipelines h′ [m] Water depth hB [m] Water depth in the area of the benching of a

sectionalised cross-section hcrit(hgr) [m] Critical depth hE [m] Energy head hf(hv) [m] Energy fall hfi(hv,E) [m] Individual energy fall, local energy fall hi(ha) [m] initial water depth hL [m] Local energy loss hM [m] Height of wave maximum in the elbow shaft hn [m] Normal water depth hNN [m] Water level referred to MSL hx [m] General flow depth ∆h [m] Difference in water level H [m] Profile height with non-circular cross-sections HCh(Hri) [m] Profile height of the flow channels of a sectionalised

cross-section

ATV-DVWK-A 110E

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Jq [1] Additional loss gradient as a result of lateral inflow Jc [1] Sole gradient with sedimentation-free operation Jcrit [1] Sole gradient at which the transport of solids ends JE [1] Energy gradient JF(JR) [1] Friction gradient JLim(Jgr) [1] Critical gradient JSo [1] Sole (bottom) gradient JW [1] Water level gradient k [mm; m] Hydraulically effective wall roughness kb [mm; m] Operational roughness kL [mm; m] Local loss coefficient (from DIN EN 752) ks [mm; m] Sand roughness kSt [m1/3/s] Coefficient according to Manning-Strickler K [m1/3/s] Velocity coefficient according to Manning-Strickler (DIN

EN 752) λ [1] Drag coefficient as a result of natural roughness λb [1] Drag coefficient as a result of operational roughness LH(Lw) [m] Length of the hydraulic jump l [m] Length (actual length) l’ [m] Length (projection on to the horizontal lP(lu) [m] Wetted perimeter lP,Ch(lu,Ri) [m] Wetted perimeter of the flow channel of a sectionalised

cross-section ∆lP*(lu*) [m] Correction value for the wetted perimeter of the flow

channel lP*

,Ch(lu*,Ri) [m] Hydraulically effective wetted perimeter of the flow channel

ν [m2/s] Kinematic viscosity m [1] Factor with inclusion of additional losses n [1] Number of partial cross-sections with sectionalised

cross-sections q [m3/(s⋅m)] Lateral inflow per unit of length in the direction of flow qe [m3/(s⋅m)] End discharge with surface runoffs Q [m3/s] Throughflow, discharge, volume flow QA(QL) [m3/s] Air component with discharge of a water-air mixture Qcrit(Qgr) [m3/s] Critical discharge QDIM [m3/s] Dimensioning discharge with water-air mixture Qe [m3/s] Throughflow at the end of a calculation stretch Qi [m3/s] Initial flow Qm [m3/s] Mean value of throughflow Qn [m3/s] Normal discharge Qs [m3/s] Throughflow at the start of a calculation stretch ∆Q [m3/s] Inflow along a calculation stretch ρ [°] Relative bending of the shaft deflection ρ [kg/m3] Density r [m] Radius rhy [m] Hydraulic radius Re [1] Reynolds Number t [s] Time co-ordinate ∆t [s] Time co-ordinate step T [°C] Temperature

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τ [N/m2] Wall transverse stress v [m/s] Average flow rate in a cross-section vc [m/s] Flow rate for sedimentation-free operation vcrit [m/s] Average flow rate at which the transport of solids ends vlim(vgr) [m/s] Limiting velocity vM(vG) [m/s] Average flow rate of a water-air mixture x [m] Path co-ordinate in the direction of flow ∆x [m] Step width of the path co-ordinate ζ [1] Loss coefficient ζC(ζR) [1] Loss coefficients for control shafts ζCS(ζVB) [1] Loss coefficients for combining structures ζfd(ζu) [1] Loss coefficients for flow diversion ζin(ζz) [1] Loss coefficient for inflow fittings ζPC(ζRV) [1] Loss coefficients for pipe connections ζPi(ζL) [1] Loss coefficient due to positional inaccuracies and

changes ζPJ(ζSt) [1] Loss coefficient for pipe joints ζS [1] Loss coefficient for special shafts ∆z [m] Bottom Indices

avail Available value of a parameter Ch Index for “flow in the area of the channel” with structured

cross-sections L Index for “flow in the area of the left benching” with

structured cross-sections max Maximum value of a parameter min Minimum value of a parameter o above R Index for “flow in the area of the right benching” with ?

cross-sections T Partial filling u Below V Complete filling Auxiliary parameters:

a1,a2,X,Y,Z

ATV-DVWK-A 110E

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Appendix

As a series of new subjects had to be dealt with and existing details had to be presented comprehensively with the revision of ATV-A 110, the working group considered somewhat broader information on the fundamentals and prerequisites to be necessary. In order not to prejudice the clarity of the standard, this is separated from the actual text and summarised in this appendix.

A1 Rules in accordance with Din EN 752, Part 4

General

(corresponds with DIN EN 752-4, Sect. 9.1.1)

For the calculation of turbulent flows in drains and sewers, the equations from Prandtl-Colebrook22) and Manning-Strickler23) are recommended.

Prandtl-Colebrook Equation

(corresponds with DIN EN 752-4, Sect. 9.1.2)

The flow rate in circular shaped, completely filled pipes is calculated in accordance with the equation:

ν⋅+

⋅⋅⋅⋅−=

)GJ2(D51.2

D71.3kloggDJ2(2v

E10E (66)

With: v the average velocity over the flow cross-section [m/s]; g acceleration due to gravity [m/s2]; D the internal diameter of the pipe [m]; JE energy gradient (energy loss per unit length) [-]; k the hydraulic roughness of the pipeline [m]; ν the kinematic viscosity of the fluid [m2/s]

For partial filling or noncircular pipes the flow rate is calculated according to Eqn. (66), in which D is replaced by 4Rh. With this, Rh is the hydraulic radius (flow face surface divided by the wetted perimeter).

Manning-Strickler Equation

(corresponds with DIN EN 752-4, Sect. 9.1.3)

The flow rate for circular shaped and noncircular shaped cross-sections is calculated using the following equation for partial and complete filling:

2/1E

3/2h JKRv = (67)

________________ 22) This equation is called the Colebrook-White Equation in the English version and Colebrook Equation in the French version. 23) This equation is called the Manning Equation in the English version and Manning-Strickler Equation in the French version.

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With: K the velocity coefficient according to Manning-Strickler [m1/3/s]; Rh the hydraulic radius [m]; JE energy gradient (energy loss per unit length) [-].

Energy losses in pipelines

(corresponds with DIN EN 752-4, Sect. 9.2.1)

The hydraulic roughness of the pipeline (k) or the velocity coefficient take into account energy losses due to pipe materials, the pipe connections and the sewer film.

Local energy losses

(corresponds with DIN EN 752-4, Sect. 9.2.2)

In addition to the energy losses described under “General”, further losses occur with combining, changes of cross-section, shafts, bends and other fittings. In case these losses are calculated directly the following equation is to be used:

g2vkh

2L

L =

with

hL the local energy loss [m], kL the loss coefficient [1], v the flow rate [m/s], g acceleration due to gravity [m/s2].

Total energy losses

(corresponds with DIN EN 752-4, Sect. 9.2.3, Paras 1 to 3)

there are two procedures for the determination of the total energy loss:

- local energy losses and the energy losses in the pipelines are added, - taking into account the local energy losses by assuming a larger hydraulic roughness

of the pipeline with the calculation of the energy losses

If recommended hydraulic roughnesses are used for pipelines or velocity coefficients, it is to be clarified whether local energy losses are taken into account in these. For a roughness k values of 0.03 mm to 3.0 m are applied, for the velocity coefficient K values from 70 m1/3s-1 to 90 m1/3s-1 Details on this are pointed out in DIN EN 752 Sect. 4 and are contained in documents listed in DIN EN Appx. 4.

If long-term deposits cannot be avoided then the anticipated energy loss as a result of reduced cross-section is to be taken into account in the calculation of the energy losses.

Velocity calculations according to Eqns. (66) and (67) can be compared approximately using the following equation:

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⋅

⋅⋅=

kD7.3log

D32g4K 10

61

(68)

with

K the velocity coefficient according to Manning-Strickler in m1/3/s, g acceleration due to gravity [m/s2], D the internal diameter of the pipe in m, k the hydraulic roughness of the pipeline in m.

Sources of additional information from other CEN countries

The informative Appendix A of DIN EN 752-4 “Drainage systems outside buildings” Part 4 “Hydraulic calculation and environmental aspects” contains all relevant details for Austria, Denmark, Finland, France, Ireland, Italy, The Netherlands, Portugal, Spain, Sweden, Switzerland and the United Kingdom.

A2 Compilation of partial filling curves; summary of profile groups; designation of profiles, which deviate from DIN 4263

The calculation of discharge processes with partial filling is extensive because, in a partially filled sewer pipe the filling height as initial value, is not known. For steady state, uniform discharge - energy line and water level profile run parallel to the sole (Fig. 2), so that filling curves can be used with which the partial filling values are based on complete filling. For partial filling conditions between h/d from some 0.8 (depending on slight differences according to profile type) up to 1.0 steady state, uniform discharge is practically impossible (QT/QV > 1.0). Therefore for this range there are no partial filling curves given. With non-uniform discharge a calculation of the water surface profile is always required, with which, naturally, the range 0.8 ≤ h/d ≤ 1.0 can be dealt with mathematically as then one is not concerned with normal water depths.

The degree of filling is without dimension. With circular pipes H = d. As the profile height, which is always measured perpendicular to the pipe axis, is a constant value independent of slope, the filling height must also be referred to the perpendicular to the pipe axis (see also Eqn. (14)).

The creation of a partial filling curve for steady state discharge processes thus is reduced to the calculation of geometrical relationship values for certain shapes of profile. In accordance with Eqns. 16 and 17, the relationship values AT/AV and rhy,T/rhy,V are required for the different degrees of filling.

The partial filling values for circular, oval and tapering cross-sections are to be taken from Figs. 13 to 15 as well as Tables 14 to 19.

For non-tabulated cross-sections the partial filling values for A and rhy = A/lP are to be recorded and evaluated analytically or using plane geometry.

The following example serves to illustrate dealing with the thus obtained partial filling curve:

Dry weather discharge: QT = 38 l/s

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Combined discharge: Qm = 1500 l/s Sole gradient: JSo = 2.5 ‰ Operational roughness: kb = 1.5 mm Sewer profile: Circular DN 1200

Required: hT and vT for dry weather and combined discharges.

Discharge quantity according to Eqns. (12) and (20):

Qv = 1897 l/s, in which vv = 1.68 m/s

Dry weather discharge:

020.01897

38QQ

v

T ==

;095.0d

hT = hT = 0.095 ⋅ 1.20 = 0.12 m

;413.0vv

v

T = vT = 0.413 ⋅ 1.68 = 0.69 m/s

Combined wastewater discharged:

v

T

v

m

QQ79.0

18971500

QQ

===

;674.0d

hT = hT = 0.674 ⋅ 1.20 = 0.81 m

;103.1vv

v

T = vT = 1.103 ⋅ 1.68 = 1.85 m/s

In Sect. 3.1.2 of the Standard it was pointed out that, for geometrically similar profile shapes, the partial filling curves can be combined. It is thus recommended, under the following listed profile shapes and their numbering (see Figs 16 and 17), to proceed as follows:

Circular profile curve for Profile Nos.: 1; 4 and 5

Oval profile curve for Profile Nos.: 2; 6; 7; 8 and 15

Tapering profile curve for Profile Nos.: 3; 9; 10; 11; 12; 13 and 14

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Fig. 13: Partial filling curve for circular sections

Fig. 14: Partial filling curves for oval sections

Fig. 15: Partial filling curves for tapering sections

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ATV-DVWK-A 110E

September 2001 65

Table 14: Partial filling values for circular cross-sections dependent on QT/QV

QT/QV vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B QT/QV vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.338 0.413 0.464 0.503 0.537

0.065 0.095 0.116 0.134 0.149

0.030 0.048 0.065 0.079 0.093

0.168 0.200 0.221 0.238 0.252

0.176 0.243 0.292 0.334 0.369

0.504 0.587 0.640 0.680 0.712

0.51 0.52 0.53 0.54 0.55

1.005 1.009 1.014 1.018 1.023

0.506 0.512 0.518 0.524 0.530

0.508 0.515 0.523 0.530 0.538

0.504 0.508 0.511 0.515 0.519

1.008 1.015 1.022 1.029 1.036

1.000 1.000 0.999 0.999 0.998

0.06 0.07 0.08 0.09 0.10

0.565 0.590 0.613 0.633 0.652

0.163 0.176 0.188 0.200 0.211

0.106 0.119 0.131 0.142 0.153

0.265 0.276 0.286 0.295 0.304

0.401 0.430 0.457 0.482 0.505

0.739 0.762 0.782 0.800 0.815

0.56 0.57 0.58 0.59 0.60

1.027 1.031 1.035 1.039 1.043

0.536 0.542 0.547 0.553 0.559

0.545 0.553 0.560 0.568 0.575

0.523 0.526 0.530 0.534 0.538

1.043 1.050 1.057 1.063 1.070

0.997 0.997 0.995 0.994 0.995

0.11 0.12 0.13 0.14 0.15

0.670 0.686 0.702 0.716 0.730

0.221 0.231 0.241 0.250 0.259

0.164 0.175 0.185 0.195 0.205

0.312 0.319 0.326 0.333 0.340

0.527 0.548 0.567 0.580 0.604

0.830 0.843 0.855 0.866 0.876

0.61 0.62 0.63 0.64 0.65

1.047 1.051 1.054 1.058 1.061

0.565 0.571 0.577 0.583 0.589

0.583 0.590 0.598 0.695 0.612

0.542 0.545 0.549 0.553 0.557

1.076 1.082 1.088 1.094 1.100

0.991 0.990 0.988 0.986 0.984

0.16 0.17 0.18 0.19 0.20

0.743 0.756 0.767 0.779 0.790

0.268 0.276 0.285 0.293 0.301

0.215 0.225 0.235 0.244 0.253

0.346 0.352 0.358 0.364 0.370

0.622 0.639 0.655 0.670 0.685

0.886 0.894 0.903 0.910 0.917

0.66 0.67 0.68 0.69 0.70

1.065 1.068 1.071 1.075 1.078

0.595 0.601 0.607 0.613 0.619

0.620 0.627 0.635 0.642 0.650

0.561 0.565 0.568 0.572 0.576

1.106 1.111 1.117 1.122 1.127

0.982 0.980 0.977 0.974 0.971

0.21 0.22 0.23 0.24 0.25

0.800 0.810 0.820 0.829 0.838

0.309 0.316 0.324 0.331 0.339

0.262 0.272 0.281 0.289 0.298

0.375 0.380 0.385 0.390 0.395

0.700 0.714 0.728 0.741 0.754

0.924 0.930 0.936 0.941 0.947

0.71 0.72 0.73 0.74 0.75

1.081 1.084 1.087 1.090 1.092

0.625 0.631 0.637 0.643 0.649

0.657 0.664 0.672 0.679 0.687

0.580 0.584 0.588 0.592 0.596

1.132 1.137 1.142 1.147 1.152

0.968 0.965 0.962 0.958 0.955

0.26 0.27 0.28 0.29 0.30

0.847 0.856 0.864 0.872 0.880

0.346 0.353 0.360 0.367 0.374

0.307 0.316 0.324 0.333 0.341

0.400 0.405 0.410 0.414 0.419

0.767 0.779 0.791 0.803 0.814

0.951 0.956 0.960 0.964 0.968

0.76 0.77 0.78 0.79 0.80

1.095 1.098 1.100 1.103 1.105

0.655 0.661 0.667 0.674 0.680

0.694 0.702 0.709 0.717 0.724

0.600 0.604 0.609 0.613 0.617

1.156 1.161 1.165 1.169 1.173

0.951 0.947 0.942 0.938 0.933

0.31 0.32 0.33 0.34 0.35

0.887 0.894 0.902 0.909 0.915

0.381 0.387 0.394 0.401 0.407

0.349 0.358 0.366 0.374 0.382

0.423 0.428 0.432 0.436 0.440

0.826 0.837 0.847 0.858 0.868

0.971 0.974 0.977 0.980 0.983

0.81 0.82 0.83 0.84 0.85

1.107 1.109 1.112 1.114 1.116

0.686 0.693 0.699 0.706 0.712

0.732 0.739 0.747 0.754 0.762

0.622 0.626 0.630 0.635 0.640

1.177 1.181 1.184 1.188 1.191

0.928 0.923 0.917 0.911 0.905

0.36 0.37 0.38 0.39 0.40

0.922 0.928 0.935 0.941 0.947

0.414 0.420 0.426 0.433 0.439

0.390 0.399 0.407 0.415 0.422

0.445 0.449 0.453 0.457 0.461

0.878 0.888 0.898 0.907 0.916

0.985 0.987 0.989 0.991 0.993

0.86 0.87 0.88 0.89 0.90

1.117 1.119 1.121 1.123 1.124

0.719 0.726 0.733 0.740 0.747

0.770 0.777 0.785 0.793 0.801

0.644 0.649 0.654 0.659 0.664

1.194 1.198 1.200 1.203 1.206

0.899 0.892 0.885 0.878 0.870

0.41 0.42 0.43 0.44 0.45

0.953 0.958 0.964 0.970 0.975

0.445 0.451 0.458 0.464 0.470

0.430 0.438 0.446 0.454 0.462

0.465 0.469 0.473 0.477 0.481

0.925 0.934 0.943 0.952 0.960

0.994 0.995 0.996 0.997 0.998

0.91 0.92 0.93 0.94 0.95

1.125 1.127 1.128 1.129 1.129

0.754 0.761 0.769 0.776 0.784

0.809 0.817 0.825 0.833 0.841

0.669 0.675 0.681 0.686 0.692

1.208 1.210 1.212 1.214 1.215

0.862 0.853 0.843 0.834 0.823

0.46 0.47 0.48 0.49 0.50

0.980 0.985 0.990 0.995 1.000

0.476 0.482 0.488 0.494 0.500

0.469 0.477 0.485 0.492 0.500

0.485 0.489 0.492 0.496 0.500

0.968 0.977 0.984 0.992 1.000

0.999 0.999 1.000 1.000 1.000

0.96 0.97 0.98 0.99 1.00

1.130 1.130 1.131 1.131 1.130

0.792 0.800 0.809 0.818 0.827

0.850 0.858 0.867 0.876 0.885

0.699 0.705 0.712 0.719 0.727

1.216 1.217 1.217 1.217 1.217

0.812 0.799 0.786 0.772 0.756

ATV-DVWK-A 110E

September 2001 66

Table 15: Partial filling values for circular cross-sections dependent on h/d

h/d vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B h/d vTvV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.1035 0.1592 0.2045 0.2440 0.2797

0.0002 0.0008 0.0018 0.0033 0.0052

0.00170.00480.00870.01340.0187

0.0638 0.0903 0.1108 0.1262 0.1436

0.0266 0.0528 0.0789 0.1047 0.1302

0.19900.28000.34120.39190.4359

0.51 0.52 0.53 0.54 0.55

1.00751.01541.02281.02991.0308

0.51670.53360.55040.56740.5843

0.5127 0.5255 0.5382 0.5509 0.5636

0.5064 0.5127 0.5191 0.5255 0.5319

1.0126 1.0248 1.0367 1.0483 1.0595

0.99980.99920.99820.99680.9950

0.06 0.07 0.08 0.09 0.10

0.3125 0.3430 0.3717 0.3989 0.4247

0.0077 0.0106 0.0139 0.0178 0.0221

0.02450.03080.03750.04460.0520

0.1575 0.1705 0.1826 0.1940 0.2048

0.1555 0.1805 0.2053 0.2298 0.2541

0.47500.51030.54260.57240.6000

0.56 0.57 0.58 0.59 0.60

1.04351.04991.05611.06201.0677

0.60130.61820.63510.65210.6689

0.5762 0.5888 0.6014 0.6140 0.6265

0.5383 0.5447 0.5511 0.5576 0.5641

1.0704 1.0811 1.0912 1.1011 1.1106

0.99280.99020.98710.98370.9798

0.11 0.12 0.13 0.14 0.15

0.4494 0.4730 0.4957 0.5175 0.5386

0.0269 0.0322 0.0379 0.0440 0.0507

0.05990.06800.07640.08510.0941

0.2152 0.2252 0.2348 0.2441 0.2532

0.2781 0.3018 0.3253 0.3485 0.3715

0.62580.64990.67260.69400.7141

0.61 0.62 0.63 0.64 0.65

1.07321.07851.08351.08831.0928

0.68570.70240.71900.73560.7519

0.6389 0.6513 0.6636 0.6759 0.6881

0.5706 0.5771 0.5837 0.5903 0.5970

1.1197 1.1285 1.1369 1.1449 1.1526

0.97550.97080.96500.96000.9539

0.16 0.17 0.18 0.19 0.20

0.5589 0.5786 0.5976 0.6161 0.6340

0.0577 0.0652 0.0732 0.0815 0.0903

0.10330.11270.12240.13230.1424

0.2620 0.2706 0.2789 0.2871 0.2952

0.3942 0.4167 0.4388 0.4607 0.4824

0.73320.75130.76840.78460.8000

0.66 0.67 0.68 0.69 0.70

1.09711.10121.10501.10861.1119

0.76820.78430.80020.81590.8313

0.7002 0.7122 0.7241 0.7360 0.7477

0.6037 0.6104 0.6172 0.6241 0.6310

1.1599 1.1667 1.1732 1.1793 1.1849

0.94740.94070.93300.92500.9165

0.21 0.22 0.23 0.24 0.25

0.6514 0.6684 0.6848 0.7008 0.7164

0.0994 0.1090 0.1190 0.1293 0.1401

0.15270.16310.17380.18450.1955

0.3031 0.3108 0.3184 0.3259 0.3333

0.5037 0.5248 0.5457 0.5662 0.5865

0.81460.82850.84170.85420.8660

0.71 0.72 0.73 0.74 0.75

1.11501.11781.12031.12261.1246

0.84660.86160.87630.89070.9048

0.7593 0.7708 0.7822 0.7934 0.8045

0.6380 0.6450 0.6522 0.6594 0.6667

1.1902 1.1950 1.1994 1.2033 1.2067

0.90750.89800.88790.87730.8660

0.26 0.27 0.28 0.29 0.30

0.7316 0.7464 0.7608 0.7748 0.7885

0.1511 0.1626 0.1744 0.1865 0.1990

0.20660.21780.22920.24070.2523

0.3406 0.3478 0.3550 0.3620 0.3690

0.6065 0.6262 0.6457 0.6649 0.6838

0.87730.88790.89800.90750.9165

0.76 0.77 0.78 0.79 0.80

1.12641.12781.12901.12991.1305

0.91850.93190.94480.95740.9695

0.8154 0.8262 0.8369 0.8473 0.8576

0.6741 0.6816 0.6892 0.6969 0.7048

1.2097 1.2123 1.2143 1.2158 1.2168

0.85420.84170.82850.81460.8000

0.31 0.32 0.33 0.34 0.35

0.8019 0.8149 0.8276 0.8400 0.8520

0.2117 0.2248 0.2382 0.2518 0.2658

0.26400.27590.28780.29980.3119

0.3759 0.3828 0.3896 0.3963 0.4043

0.7024 0.7207 0.7387 0.7565 0.7740

0.92500.93300.94040.94740.9539

0.81 0.82 0.83 0.84 0.85

1.13071.1306

0.98110.9922

0.8677 0.8776 0.8873 0.8967 0.9059

0.7129 0.7211 0.7294 0.7380 0.7468

1.2172 1.2171 1.2164 1.2150 1.2131

0.78460.76840.75130.73320.7141

0.36 0.37 0.38 0.39 0.40

0.8638 0.8753 0.8865 0.8974 0.9080

0.2800 0.2944 0.3091 0.3240 0.3392

0.32410.33640.34870.36110.3735

0.4097 0.4163 0.4229 0.4294 0.4359

0.7911 0.8080 0.8246 0.8409 0.8569

0.96000.96560.97080.97550.9798

0.86 0.87 0.88 0.89 0.90

0.9149 0.9236 0.9320 0.9401 0.9480

0.7559 0.7652 0.7748 0.7848 0.7952

1.2104 1.2071 1.2029 1.1980 1.1921

0.69400.67260.64990.62580.6000

0.41 0.42 0.43 0.44 0.45

0.9184 0.9284 0.9383 0.9478 0.9572

0.3545 0.3701 0.3858 0.4017 0.4177

0.38600.39860.41120.42380.4364

0.4424 0.4489 0.4553 0.4617 0.4681

0.8726 0.8880 0.9031 0.9179 0.9323

0.98370.98710.99020.99280.9950

0.91 0.92 0.93 0.94 0.95

0.9554 0.9625 0.9692 0.9755 0.9813

0.8060 0.8174 0.8295 0.8425 0.8564

1.1853 1.1775 1.1684 1.1579 1.1458

0.57240.54260.51030.47500.4359

0.46 0.47 0.48 0.49 0.50

0.9662 0.9750 0.9836 0.9919 1.0000

0.4340 0.4503 0.4668 0.4833 0.5000

0.44910.46180.47450.48730.5000

0.4745 0.4809 0.4873 0.4936 0.5000

0.9465 0.9604 0.9739 0.9871 1.0000

0.99680.99820.99920.99981.0000

0.96 0.97 0.98 0.99 1.00

1.0000

1.0000

0.9866 0.9913 0.9952 0.9983 1.0000

0.8718 0.8892 0.9097 0.9362 1.0000

1.1316 1.1148 1.0941 1.0663 1.0000

0.39190.34120.28000.19900.0000

ATV-DVWK-A 110E

September 2001 67

Table 16: Partial filling values for oval cross-sections dependent on QT/QV

QT/QV vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B QT/QV v‘/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.385 0.463 0.514 0.553 0.584

0.070 0.100 0.123 0.143 0.161

0.026 0.043 0.058 0.072 0.086

0.120 0.148 0.169 0.187 0.203

0.217 0.292 0.345 0.387 0.423

0.407 0.471 0.518 0.556 0.588

0.51 0.52 0.53 0.54 0.55

0.993 0.997 1.001 1.005 1.008

0.555 0.562 0.568 0.574 0.580

0.514 0.522 0.530 0.538 0.545

0.519 0.524 0.529 0.534 0.538

0.989 0.995 1.001 1.007 1.013

0.981 0.983 0.985 0.987 0.989

0.06 0.07 0.08 0.09 0.10

0.610 0.633 0.654 0.672 0.689

0.177 0.192 0.206 0.219 0.231

0.098 0.111 0.122 0.134 0.145

0.217 0.230 0.241 0.253 0.263

0.454 0.481 0.507 0.530 0.551

0.616 0.640 0.662 0.682 0.701

0.56 0.57 0.58 0.59 0.60

1.012 1.016 1.019 1.023 1.026

0.586 0.592 0.598 0.604 0.610

0.553 0.561 0.569 0.577 0.585

0.543 0.548 0.552 0.557 0.561

1.019 1.025 1.031 1.036 1.042

0.990 0.992 0.993 0.995 0.995

0.11 0.12 0.13 0.14 0.15

0.705 0.720 0.733 0.746 0.758

0.243 0.255 0.265 0.276 0.286

0.156 0.167 0.177 0.188 0.198

0.273 0.282 0.291 0.300 0.308

0.572 0.591 0.608 0.625 0.642

0.718 0.733 0.748 0.762 0.774

0.61 0.62 0.63 0.64 0.65

1.029 1.033 1.036 1.039 1.042

0.616 0.622 0.628 0.634 0.640

0.593 0.600 0.608 0.616 0.624

0.566 0.570 0.575 0.579 0.584

1.047 1.053 1.058 1.063 1.068

0.996 0.997 0.998 0.998 0.999

0.16 0.17 0.18 0.19 0.20

0.769 0.780 0.790 0.800 0.809

0.296 0.306 0.315 0.324 0.333

0.208 0.218 0.228 0.237 0.247

0.317 0.324 0.332 0.399 0.346

0.657 0.672 0.686 0.700 0.713

0.786 0.798 0.808 0.819 0.828

0.66 0.67 0.68 0.69 0.70

1.045 1.048 1.051 1.054 1.057

0.646 0.652 0.658 0.664 0.670

0.631 0.639 0.647 0.655 0.662

0.588 0.593 0.597 0.602 0.606

1.073 1.078 1.083 1.088 1.092

0.999 1.000 1.000 1.000 1.000

0.21 0.22 0.23 0.24 0.25

0.818 0.827 0.836 0.844 0.851

0.342 0.350 0.359 0.367 0.375

0.257 0.266 0.275 0.285 0.294

0.353 0.360 0.367 0.374 0.380

0.726 0.738 0.750 0.762 0.773

0.837 0.846 0.854 0.862 0.870

0.71 0.72 0.73 0.74 0.75

1.060 1.062 1.065 1.068 1.070

0.676 0.682 0.688 0.693 0.699

0.670 0.678 0.685 0.693 0.701

0.611 0.615 0.620 0.624 0.629

1.097 1.102 1.106 1.110 1.115

1.000 0.999 0.998 0.997 0.995

0.26 0.27 0.28 0.29 0.30

0.859 0.866 0.873 0.880 0.887

0.383 0.391 0.399 0.406 0.414

0.303 0.312 0.321 0.330 0.338

0.386 0.392 0.398 0.404 0.410

0.784 0.794 0.805 0.815 0.825

0.877 0.884 0.890 0.896 0.902

0.76 0.77 0.78 0.79 0.80

1.073 1.075 1.078 1.080 1.082

0.705 0.711 0.717 0.723 0.729

0.708 0.716 0.724 0.731 0.739

0.633 0.638 0.642 0.647 0.651

1.119 1.123 1.127 1.131 1.135

0.993 0.991 0.988 0.986 0.982

0.31 0.32 0.33 0.34 0.35

0.893 0.899 0.905 0.911 0.917

0.421 0.428 0.436 0.443 0.450

0.347 0.356 0.365 0.373 0.382

0.416 0.422 0.427 0.433 0.439

0.834 0.844 0.853 0.862 0.870

0.908 0.914 0.919 0.924 0.929

0.81 0.82 0.83 0.84 0.85

1.084 1.087 1.089 1.091 1.093

0.735 0.741 0.747 0.753 0.760

0.747 0.755 0.762 0.770 0.778

0.656 0.661 0.665 0.670 0.675

1.139 1.142 1.146 1.149 1.152

0.979 0.975 0.970 0.966 0.960

0.36 0.37 0.38 0.39 0.40

0.923 0.928 0.933 0.939 0.944

0.457 0.464 0.471 0.478 0.484

0.390 0.399 0.407 0.416 0.424

0.444 0.449 0.455 0.460 0.465

0.879 0.887 0.896 0.904 0.911

0.933 0.938 0.942 0.946 0.950

0.86 0.87 0.88 0.89 0.90

1.094 1.096 1.098 1.100 1.101

0.766 0.772 0.779 0.785 0.792

0.786 0.794 0.802 0.809 0.817

0.680 0.685 0.690 0.695 0.701

1.155 1.158 1.161 1.164 1.166

0.955 0.949 0.942 0.935 0.927

0.41 0.42 0.43 0.44 0.45

0.949 0.954 0.955 0.963 0.968

0.491 0.498 0.504 0.511 0.517

0.432 0.440 0.449 0.457 0.465

0.470 0.475 0.480 0.485 0.490

0.919 0.927 0.934 0.941 0.949

0.953 0.957 0.960 0.963 0.966

0.91 0.92 0.93 0.94 0.95

1.102 1.104 1.105 1.106 1.107

0.798 0.805 0.812 0.819 0.826

0.825 0.834 0.842 0.850 0.858

0.706 0.712 0.718 0.724 0.730

1.169 1.171 1.173 1.175 1.176

0.919 0.910 0.900 0.889 0.878

0.46 0.47 0.48 0.49 0.50

0.972 0.976 0.981 0.985 0.989

0.524 0.530 0.536 0.543 0.549

0.473 0.481 0.489 0.498 0.506

0.495 0.500 0.505 0.510 0.515

0.956 0.962 0.969 0.976 0.982

0.969 0.972 0.974 0.977 0.979

0.96 0.97 0.98 0.99 1.00

1.107 1.108 1.108 1.108 1.108

0.834 0.842 0.850 0.858 0.867

0.867 0.876 0.884 0.893 0.903

0.736 0.743 0.750 0.758 0.766

1.177 1.178 1.179 1.179 1.178

0.865 0.851 0.836 0.819 0.800

ATV-DVWK-A 110E

September 2001 68

Table 17: Partial filling values for oval cross-sections dependent on h/H

h/H vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B h/H vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.1209 0.1848 0.2359 0.2797 0.3185

0.0002 0.0008 0.0018 0.0033 0.0051

0.00150.00420.00760.01160.0161

0.0439 0.0624 0.0768 0.0892 0.1003

0.0340 0.0671 0.0992 0.1302 0.1603

0.17060.23750.28620.32500.3571

0.51 0.52 0.53 0.54 0.55

0.96240.96940.97630.98300.9895

0.43890.43430.46990.48560.5015

0.4560 0.4683 0.4813 0.4940 0.5068

0.4848 0.4925 0.5001 0.5077 0.5154

0.9406 0.9516 0.9623 0.9729 0.9833

0.96300.96760.97190.97580.9795

0.06 0.07 0.08 0.09 0.10

0.3534 0.3852 0.4139 0.4400 0.4641

0.0074 0.0101 0.0131 0.0164 0.0201

0.02090.02610.03160.03730.0433

0.1105 0.1201 0.1295 0.1388 0.1480

0.1894 0.2173 0.2438 0.2689 0.2928

0.38420.40750.42950.45090.4718

0.56 0.57 0.58 0.59 0.60

0.99591.00221.00881.01421.0201

0.51750.53360.54980.56620.5827

0.5196 0.5324 0.5453 0.5583 0.5712

0.5230 0.5306 0.5382 0.5458 0.5533

0.9935 1.0035 1.0133 1.0229 1.0323

0.98290.98590.98870.99120.9933

0.11 0.12 0.13 0.14 0.15

0.4866 0.5077 0.5277 0.5467 0.5648

0.0242 0.0285 0.0333 0.0383 0.0437

0.04960.05620.06300.07010.0774

0.1571 0.1662 0.1752 0.1841 0.1930

0.3159 0.3381 0.3596 0.3805 0.4009

0.49220.51210.53140.55020.5686

0.61 0.62 0.63 0.64 0.65

1.02571.03141.03671.04191.0470

0.59920.61590.63260.64940.6662

0.5842 0.5972 0.6102 0.6233 0.6363

0.5609 0.5685 0.5761 0.5836 0.5912

1.0415 1.0505 1.0593 1.0679 1.0763

0.99520.99670.99800.99890.9996

0.16 0.17 0.18 0.19 0.20

0.5821 0.5988 0.6149 0.6303 0.6453

0.0494 0.0555 0.0619 0.0686 0.0757

0.08490.09270.10070.10890.1173

0.2018 0.2106 0.2193 0.2279 0.2365

0.4208 0.4402 0.4592 0.4779 0.4962

0.58640.60380.62080.63730.6533

0.66 0.67 0.68 0.69 0.70

1.05201.05691.06161.06611.0705

0.68320.70010.71710.73410.7510

0.6494 0.6624 0.6755 0.6885 0.7015

0.5988 0.6064 0.6139 0.6215 0.6291

1.0845 1.0925 1.1003 1.1079 1.1152

0.99990.99990.99920.99750.9950

0.21 0.22 0.23 0.24 0.25

0.6598 0.6739 0.6876 0.70090.7138

0.0831 0.0908 0.0989 0.1073 0.1160

0.12600.13480.14380.15310.1625

0.2450 0.2535 0.2619 0.2703 0.2786

0.5142 0.5318 0.5492 0.5662 0.5830

0.66890.68410.69890.71320.7272

0.71 0.72 0.73 0.74 0.75

1.07471.07871.08261.08621.0896

0.76790.78470.80140.81800.8344

0.7145 0.7274 0.7403 0.7531 0.7658

0.6367 0.6444 0.6520 0.6598 0.6676

1.1222 1.1289 1.1354 1.1414 1.1471

0.99150.98710.98180.97550.9682

0.26 0.27 0.28 0.29 0.30

0.7264 0.7386 0.7506 0.7623 0.7737

0.1250 0.1343 0.1439 0.1538 0.1641

0.17210.18180.19170.20180.2121

0.2870 0.2952 0.3034 0.3116 0.3198

0.5996 0.6159 0.6319 0.6477 0.6632

0.74070.75390.76670.77900.7911

0.76 0.77 0.78 0.79 0.80

1.09271.09561.09831.10061.1027

0.85060.86650.88210.89750.9125

0.7784 0.7908 0.8032 0.8154 0.8275

0.6754 0.6833 0.6913 0.6994 0.7076

1.1525 1.1574 1.1618 1.1658 1.1694

0.96000.95070.94040.92900.9165

0.31 0.32 0.33 0.34 0.35

0.7848 0.7956 0.8063 0.8166 0.8268

0.1746 0.1854 0.1965 0.2079 0.2196

0.22250.23310.24380.25460.2656

0.3279 0.3360 0.3440 0.3520 0.3600

0.6786 0.6937 0.7085 0.7232 0.7376

0.80270.81400.82490.83540.8456

0.81 0.82 0.83 0.84 0.85

1.10451.10601.10711.10781.1082

0.92710.94120.95490.96800.9806

0.8393 0.8510 0.8625 0.8738 0.8848

0.7159 0.7244 0.7330 0.7418 0.7507

1.1724 1.1748 1.1767 1.1780 1.1787

0.90280.88790.87170.85420.8352

0.36 0.37 0.38 0.39 0.40

0.8367 0.8464 0.8559 0.8652 0.8743

0.2315 0.2437 0.2562 0.2689 0.2818

0.27670.28790.29930.31070.3332

0.3680 0.3759 0.3838 0.3917 0.3996

0.7519 0.7659 0.7797 0.7933 0.8067

0.85540.86490.87410.88290.8914

0.86 0.87 0.88 0.89 0.90

1.1082 0.9925 0.8956 0.9061 0.9263 0.9163 0.9357

0.7599 0.7693 0.7790 0.7890 0.7994

1.1786 1.1778 1.1763 1.1738 1.1705

0.81460.79240.76840.74240.7141

0.41 0.42 0.43 0.44 0.45

0.8832 0.8920 0.9005 0.9088 0.9170

0.2950 0.3085 0.3221 0.3360 0.3501

0.33400.34580.35770.36970.3818

0.4074 0.4152 0.4230 0.4308 0.4386

0.8198 0.8328 0.8456 0.8582 0.8705

0.89950.90730.91480.92190.9287

0.91 0.92 0.93 0.94 0.95

0.9448 0.9535 0.9618 0.9695 0.9767

0.8102 0.8216 0.8335 0.8463 0.8601

1.1661 1.1606 1.1538 1.1456 1.1356

0.68340.64990.61310.57240.5268

0.46 0.47 0.48 0.49 0.50

0.9250 0.9328 0.9405 0.9480 0.9553

0.3644 0.3789 0.3936 0.4085 0.4236

0.39400.40620.41860.43100.4435

0.4463 0.4540 0.4618 0.4695 0.4771

0.8827 0.8947 0.9065 0.9180 0.9294

0.93520.94140.94730.95280.9580

0.96 0.97 0.98 0.99 1.00

1.0000

1.0000

0.9833 0.9891 0.9940 0.9979 1.0000

0.8752 0.8922 0.9122 0.9381 1.0000

1.1235 1.1086 1.0897 1.0638 1.0000

0.47500.41460.34120.24310.0000

ATV-DVWK-A 110E

September 2001 69

Table 18: Partial filling values for tapering cross-sections dependent on QT/QV

QT/QV vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B QT/QV vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.304 0.372 0.419 0.458 0.491

0.064 0.089 0.107 0.123 0.136

0.033 0.054 0.072 0.087 0.102

0.222 0.262 0.288 0.305 0.318

0.149 0.205 0.248 0.286 0.320

0.611 0.717 0.786 0.827 0.856

0.51 0.52 0.53 0.54 0.55

0.998 1.003 1.007 1.012 1.017

0.471 0.477 0.483 0.489 0.495

0.511 0.519 0.526 0.533 0.541

0.513 0.517 0.520 0.523 0.526

0.996 1.004 1.012 1.020 1.027

0.979 0.977 0.975 0.973 0.970

0.06 0.07 0.08 0.09 0.10

0.520 0.546 0.570 0.592 0.612

0.149 0.160 0.171 0.181 0.191

0.115 0.128 0.140 0.152 0.163

0.328 0.337 0.345 0.352 0.358

0.351 0.380 0.407 0.432 0.456

0.878 0.896 0.910 0.933 0.933

0.56 0.57 0.58 0.59 0.60

1.022 1.028 1.030 1.035 1.039

0.501 0.507 0.513 0.519 0.525

0.548 0.556 0.563 0.570 0.577

0.530 0.533 0.536 0.540 0.543

1.035 1.042 1.049 1.056 1.063

0.968 0.966 0.963 0.961 0.958

0.11 0.12 0.13 0.14 0.15

0.631 0.649 0.665 0.681 0.696

0.200 0.209 0.218 0.226 0.234

0.174 0.185 0.195 0.206 0.215

0.364 0.370 0.375 0.380 0.385

0.479 0.500 0.521 0.541 0.560

0.942 0.950 0.957 0.964 0.969

0.61 0.62 0.63 0.64 0.65

1.043 1.047 1.051 1.055 1.059

0.531 0.537 0.543 0.549 0.555

0.585 0.592 0.599 0.607 0.614

0.547 0.550 0.553 0.557 0.560

1.070 1.077 1.083 1.089 1.096

0.655 0.952 0.949 0.946 0.943

0.16 0.17 0.18 0.19 0.20

0.710 0.724 0.737 0.749 0.761

0.242 0.250 0.257 0.265 0.272

0.225 0.235 0.244 0.254 0.263

0.389 0.394 0.398 0.402 0.406

0.579 0.596 0.614 0.630 0.647

0.974 0.978 0.982 0.986 0.989

0.66 0.67 0.68 0.69 0.70

1.062 1.066 1.070 1.073 1.076

0.561 0.567 0.574 0.580 0.586

0.621 0.628 0.636 0.643 0.650

0.564 0.567 0.571 0.574 0.578

1.102 1.108 1.114 1.119 1.125

0.940 0.936 0.933 0.929 0.925

0.21 0.22 0.23 0.24 0.25

0.773 0.784 0.795 0.805 0.815

0.280 0.287 0.294 0.301 0.307

0.272 0.281 0.289 0.298 0.307

0.410 0.414 0.418 0.422 0.425

0.662 0.678 0.692 0.707 0.721

0.991 0.993 0.995 0.997 0.998

0.71 0.72 0.73 0.74 0.75

1.080 1.083 1.086 1.089 1.092

0.592 0.598 0.605 0.611 0.618

0.658 0.665 0.672 0.679 0.687

0.582 0.585 0.589 0.593 0.596

1.131 1.136 1.141 1.146 1.151

0.922 0.918 0.913 0.909 0.905

0.26 0.27 0.28 0.29 0.30

0.825 0.834 0.843 0.852 0.861

0.314 0.321 0.327 0.334 0.341

0.315 0.324 0.332 0.340 0.349

0.429 0.433 0.436 0.440 0.443

0.735 0.748 0.761 0.774 0.787

0.999 1.000 1.000 1.000 1.000

0.76 0.77 0.78 0.79 0.80

1.095 1.098 1.101 1.103 1.106

0.624 0.630 0.637 0.644 0.650

0.694 0.701 0.709 0.716 0.724

0.600 0.604 0.608 0.612 0.616

1.156 1.161 1.166 1.170 1.174

0.900 0.895 0.890 0.885 0.880

0.31 0.32 0.33 0.34 0.35

0.869 0.877 0.885 0.393 0.900

0.347 0.354 0.360 0.366 0.373

0.357 0.365 0.373 0.381 0.389

0.447 0.450 0.454 0.457 0.460

0.799 0.811 0.822 0.834 0.845

1.000 1.000 0.999 0.999 0.998

0.81 0.82 0.83 0.84 0.85

1.108 1.111 1.113 1.115 1.117

0.657 0.664 0.670 0.677 0.684

0.731 0.738 0.746 0.753 0.761

0.620 0.624 0.629 0.633 0.637

1.179 1.183 1.187 1.190 1.194

0.874 0.869 0.863 0.857 0.850

0.36 0.37 0.38 0.39 0.40

0.907 0.914 0.921 0.928 0.935

0.379 0.385 0.391 0.398 0.404

0.397 0.405 0.412 0.420 0.428

0.464 0.467 0.470 0.474 0.477

0.856 0.866 0.877 0.887 0.897

0.998 0.997 0.996 0.995 0.994

0.86 0.87 0.88 0.89 0.90

1.119 1.121 1.123 1.125 1.126

0.691 0.699 0.706 0.713 0.721

0.768 0.776 0.784 0.791 0.799

0.642 0.646 0.651 0.656 0.661

1.197 1.201 1.204 1.207 1.210

0.844 0.837 0.829 0.822 0.814

0.41 0.42 0.43 0.44 0.45

0.941 0.947 0.953 0.959 0.965

0.410 0.416 0.422 0.428 0.434

0.436 0.443 0.451 0.459 0.466

0.480 0.484 0.487 0.490 0.494

0.907 0.917 0.926 0.936 0.945

0.993 0.992 0.991 0.990 0.988

0.91 0.92 0.93 0.94 0.95

1.128 1.129 1.130 1.131 1.132

0.728 0.736 0.744 0.752 0.761

0.807 0.815 0.825 0.831 0.839

0.666 0.671 0.676 0.682 0.688

1.212 1.214 1.217 1.218 1.220

0.805 0.797 0.788 0.778 0.767

0.46 0.47 0.48 0.49 0.50

0.971 0.976 0.982 0.987 0.992

0.440 0.446 0.452 0.458 0.464

0.474 0.481 0.489 0.496 0.504

0.497 0.500 0.503 0.507 0.510

0.954 0.963 0.971 0.980 0.988

0.987 0.986 0.984 0.982 0.980

0.96 0.97 0.98 0.99 1.00

1.133 1.134 1.134 1.134 1.134

0.769 0.778 0.787 0.797 0.807

0.847 0.856 0.864 0.873 0.882

0.694 0.700 0.707 0.714 0.721

1.221 1.222 1.223 1.223 1.223

0.757 0.745 0.732 0.719 0.704

ATV-DVWK-A 110E

September 2001 70

Table 19: Partial filling values for tapering cross-sections dependent on h/H

h/H vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B h/H vT/vV h/d AT/AV lP,T/lP,V rhy,T/rhy,V bT/B

0.01 0.02 0.03 0.04 0.05

0.0961 0.1479 0.1903 0.2275 0.2613

0.0002 0.0009 0.0020 0.0037 0.0060

0.00210.00580.01070.01640.0229

0.0875 0.1238 0.1517 0.1753 0.1961

0.0236 0.0470 0.0703 0.0936 0.1168

0.24450.34510.42190.48620.5426

0.51 0.52 0.53 0.54 0.55

1.02851.03571.04261.04921.0556

0.57560.59220.60880.62520.6417

0.5597 0.5718 0.5839 0.5959 0.6079

0.5350 0.5406 0.5462 0.5518 0.5574

1.0460 1.0577 1.0690 1.0799 1.0905

0.96420.96000.95550.95070.9457

0.06 0.07 0.08 0.09 0.10

0.2925 0.3217 0.3493 0.3756 0.4007

0.0088 0.0122 0.0161 0.0207 0.0258

0.03010.03790.04620.05510.0644

0.2150 0.2324 0.2486 0.2638 0.2783

0.1399 0.1629 0.1859 0.2087 0.2315

0.59320.63950.68230.72230.7599

0.56 0.57 0.58 0.59 0.60

1.06171.06761.07321.07801.0837

0.65800.67430.69050.70650.7224

0.6198 0.6316 0.6434 0.6550 0.6666

0.5631 0.5688 0.5746 0.5804 0.5862

1.1006 1.1103 1.1197 1.1287 1.1373

0.94040.93490.92290.91650.9165

0.11 0.12 0.13 0.14 0.15

0.4254 0.4504 0.4752 0.4994 0.5230

0.0316 0.0380 0.0451 0.0528 0.0610

0.07420.08440.09490.10570.1167

0.2914 0.3024 0.3121 0.3209 0.3291

0.2547 0.2791 0.3041 0.3293 0.3545

0.79350.82050.84320.86270.8800

0.61 0.62 0.63 0.64 0.65

1.08861.09321.09761.10181.1057

0.73820.75390.76940.78470.7998

0.6782 0.6896 0.7009 0.7122 0.7233

0.5920 0.5979 0.6038 0.6099 0.6160

1.1455 1.1533 1.1607 1.1677 1.1743

0.90980.90280.89550.88790.8800

0.16 0.17 0.18 0.19 0.20

0.5460 0.5682 0.5899 0.6108 0.6312

0.0698 0.0791 0.0889 0.0993 0.1100

0.12790.13920.15080.16250.1743

0.3367 0.3440 0.3509 0.3575 0.3640

0.3797 0.4048 0.4297 0.4544 0.4789

0.89520.90890.92120.93220.9422

0.66 0.67 0.68 0.69 0.70

1.10931.11271.11591.11881.1215

0.81470.82940.84380.85800.8720

0.7344 0.7453 0.7562 0.7669 0.7775

0.6221 0.6283 0.6345 0.6408 0.6472

1.1806 1.1864 1.1918 1.1968 1.2014

0.87170.86310.85420.84490.8352

0.21 0.22 0.23 0.24 0.25

0.6509 0.6701 0.6887 0.7067 0.7242

0.1212 0.1329 0.1449 0.1574 0.1702

0.18630.19830.21050.22270.2350

0.3702 0.3763 0.3822 0.3881 0.3938

0.5031 0.5270 0.5506 0.5739 0.5968

0.95120.95930.96650.97290.9786

0.71 0.72 0.73 0.74 0.75

1.12391.12611.12801.12971.1311

0.88560.89900.91200.92480.9371

0.7880 0.7983 0.8085 0.8186 0.8285

0.6536 0.6601 0.6668 0.6735 0.6803

1.2055 1.2093 1.2126 1.2155 1.2179

0.82510.81460.80370.79240.7806

0.26 0.27 0.28 0.29 0.30

0.7412 0.7577 0.7737 0.7892 0.8043

0.1834 0.1969 0.2107 0.2248 0.2397

0.24740.25980.27230.28480.2974

0.3994 0.4050 0.4105 0.4160 0.4214

0.6193 0.6415 0.6633 0.6848 0.7058

0.98350.98780.99140.99430.9967

0.76 0.77 0.78 0.79 0.80

1.13231.13311.13371.13401.1341

0.94920.96080.97200.98280.9931

0.8383 0.8479 0.8573 0.8666 0.8757

0.6872 0.6942 0.7014 0.7086 0.7161

1.2199 1.2214 1.2224 1.2229 1.2230

0.76840.75560.74240.72850.7141

0.31 0.32 0.33 0.34 0.35

0.8189 0.8330 0.8465 0.8599 0.8726

0.2538 0.2687 0.2838 0.2991 0.3145

0.31000.32260.33520.34780.3604

0.4268 0.4321 0.4375 0.4428 0.4482

0.7263 0.7465 0.7662 0.7854 0.8042

0.99840.99951.00000.99990.9997

0.81 0.82 0.83 0.84 0.85

0.8846 0.8934 0.9012 0.9102 0.9183

0.7236 0.7314 0.7393 0.7475 0.7558

1.2225 1.2215 1.2199 1.2177 1.2150

0.69910.68340.66710.64990.6320

0.36 0.37 0.38 0.39 0.40

0.8850 0.8970 0.9035 0.9197 0.9306

0.3301 0.3459 0.3618 0.3778 0.3940

0.37300.38560.39820.41080.4234

0.4536 0.4589 0.4643 0.4697 0.4750

0.8225 0.8403 0.8577 0.8747 0.8912

0.99920.99850.99750.99640.9950

0.86 0.87 0.88 0.89 0.90

0.9261 0.9337 0.9411 0.9482 0.9550

0.7644 0.7733 0.7825 0.7920 0.8020

1.2116 1.2075 1.2027 1.1972 1.1908

0.61310.59330.57240.55030.5268

0.41 0.42 0.43 0.44 0.45

0.9410 0.9512 0.9610 0.9705 0.9797

0.4102 0.4265 0.4430 0.4594 0.4760

0.43590.44840.46090.47340.4858

0.4804 0.4858 0.4912 0.4966 0.5021

0.9074 0.9231 0.9383 0.9532 0.9677

0.99340.99150.98940.98710.9846

0.91 0.92 0.93 0.94 0.95

0.9615 0.9676 0.9734 0.9789 0.9839

0.8124 0.8233 0.8350 0.8474 0.8609

1.1835 1.1753 1.1659 1.1552 1.1429

0.50810.47500.44610.41460.3800

0.46 0.47 0.48 0.49 0.50

0.9885 0.9971 1.0054 1.0134 1.0211

0.4925 0.5091 0.5258 0.5424 0.5590

0.49820.51060.52290.53520.5475

0.5075 0.5130 0.5185 0.5240 0.5295

0.9817 0.9954 1.0086 1.0215 1.0340

0.98180.97880.97550.97200.9682

0.96 0.97 0.98 0.99 1.00

1.0000

1.0000

0.9885 0.9925 0.9959 0.9985 1.0000

0.8757 0.8925 0.9123 0.9381 1.0000

1.1287 1.1120 1.0916 1.0644 1.0000

0.34120.29660.24310.17260.0000

ATV-DVWK-A 110E

September 2001 71

A3 Calculation of backwater curves, water surface profile curves

In Sect. 3.2.1.3 it has already been explained that the discharge in open channels, as a rule, takes place unevenly, that is for the flow depth the following applies h = f(x) which is to be determined via Eqn. 23.

Possible water surface curves are sketched in Fig. 18 under the assumption that hn ≤ 0.8d for subcritical and supercritical normal outflow. The water surface profiles marked respectively by the figures 1 to 3 have the following significance:

Fig. 18: Water surface lines - principle paths

(hn > hcrit) subcritical normal outflow (1) backwater curve with h(x) ≥ hn (2) depression curve (hcrit) ≤ h(x) ≤ hn(3) retarded spill stream h(x) < hcrit with subsequent hydraulic jump

(hn < hcrit) supercritical normal outflow (1) backwater curve with h(x) > hcrit (2) depression curve (hcrit) ≥ h(x) ≥ hn(3) retarded spill stream h(x) ≤ hn

A4 Enlargement factor fair

The dimension fAir, required in Sect. 7 - Steep Stretches - Eqn (55), for the determination of the air transfer with steep stretches, dependent on sole gradient diameter and operational roughness is developed and published24) by P. Volkart, Zürich. The diagrams required for the application of the standard are given below; interpolation can be applied for deviating kb values.

An example calculation is given in 25).

_____________ 24) Volkart, P.: Hydraulische Bemessung steiler Kanalisationsleitungen unter Berücksichtigung der Luftaufnahme [Hydraulic

dimensioning of steep drains taking into account air take-up] 25) Volkart, P.: Hydraulische Bemessung teigefüllter Steileitungen [Hydraulic dimensioning of partially filled steep pipelines]

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Fig. 19: Enlargement factor fAir for kb = 0.1 mm

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Fig. 20: Enlargement factor fAir for kb = 0.4 mm

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Fig. 21: Enlargement factor fAir for kb = 1.0 mm

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Fig. 22: Enlargement factor fAir for kb = 1.5 mm

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A5 Section of a λ-Re Diagram (Moody Diagram)

During the preparation of this standard it was considered to be sensible and necessary to enlarge the Moody Diagram (see Fig.1) and to reproduce more values for the parameter d/k. This is carried out for the range

105 ≤ Re ≤ 107

Fig. 23: Moody Diagram (enlarged section)

A6 Sand roughness, ripple roughness

The employment of the Prandtl-Colebrook equation for the determination of friction loss assumes technically rough wall conditions, i.e. a statistically seen random distribution of roughness investigations on the pipe wall. Diverging forms of roughness produce deviating friction losses.

In the case of the sand roughness (according to Nikuradse) smaller losses occur in the transition range; in the case of ripple roughness, that is systematic roughness occurring transversely to the direction of flow, in part significantly higher losses occur. With regard to the mathematical approaches and applicability attention is drawn to the literature26) 27). _________________ 26) Schröder, R and Knauf, D.: Über das hydraulische Widerstandsverhalten von Beton- und Stahlbetonrohren im Übergangsbereich

[On the hydraulic resistance behaviour of concrete and reinforced concrete in the transition zone 27) Schröder R.C.M.: Hydraulische Methoden zur Erfassung von Rauheiten, [Hydraulic methods for the recording of roughness],

DVWK Publications Series, No. 92, Paul-Parey-Verlag, Hamburg and Berlin 1990

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A7 Linkage calculation Prandtl-Colebrook/Manning-Strickler

One flow formula still employed today is stated by Manning and later confirmed experimentally by Strickler. As proven by28), Eqn. 19 according to Manning-Strickler is, in certain ranges, almost identical with the Prandtl-Colebrook formula (Eqn. 13). Namely, if the discharge is in the completely rough zone, equivalent to

k > 30 υ (Q ⋅ (g ⋅ JE)2)-1/5

and in the range of relevant roughness of

2000 > d/k > 20

then the deviations in the throughflow Q between the two Eqns. 19 and 13 are smaller than ± 5 %. If both the above given criteria are met then, with the aid of the formula

2.8g

kk2/1

6/1St =

⋅

a direct relationship between the hydraulically effective wall roughness k according to Prandtl-Colebrook and the coefficient kSt according to Manning-Strickler is created. It should be noted that both k as well as kSt only define the wall roughness and are independent of the other flow parameters. Table 20 shows the relationship between k and kSt.

A further approximation for the conversion of kST to k and vice versa is given in DIN EN 752 Part 4 under Sect. 9.2.3 29) as

⋅

⋅

⋅=

kd71.3lg

d32g4k

6/1

St (69)

The approximation formula also directly covers the influence of the relative roughness d/k.

Table 20: Relationship between k and kSt

k [mm] kSt [m1/3/s]

Range of d or 4rhy [mm]

0.1 119

>200

0.25 102

>500

0.5 91

>1000

0.75 85

>1500

1.0 81

>2000

1.5 76

>3000

2.0 72

>4000

k [mm] kSt [m1/3/s]

Range of d or 4rhy [mm]

5.0 62

>100

10.5 55

>200

20.0 49

>400

50.0 42

>1000

100.0 38

>2000

_____________ 28) Hager, W.H.: Fließformeln für turbulente Strömungen [Flow formulas for turbulent flow] 29) The appropriate Equation 4 in DIN EN 752-4 contains a printing error which has been rectified here in Eqn (69)

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A8 Example calculation for the individual concept

Given

Impounded combined wastewater sewer DN 500 (h/d > 1.0) with special shafts. Section length L = 61 m, of which one shaft = 1 m Individual pipe length l = 1.25 m (old stock) Per section 10 inflow fittings din = 150 mm

Sought

kb (mm) (Note: in accordance with Sect. 4.3 b) due to impounding kb cannot be taken from Table 4).

Loss coefficients

1. Loss coefficient ζL according to Table 5 (Sect. 4.2.1) ζL for DN 500 = 0.1

2. Loss coefficient ζPC according to Table 6 (Sect. 4.2.2) ζPC for DN 500 = 0.003

3. Loss coefficient ζi according to Table 7 (Sect. 4.2.3) ζi for din/H = 150/500 = 0.3 : 0.004

4. Loss coefficient ζfd,S according to Table 9 (Sect. 4.2.5) ζfd,S for impounded shafts without diversion: 0.85

Calculation

a) Applying k = 0.1 mm for the effective wall roughness and v = 0.8 m/s for the flow rate (comp. 4.3), λ from Eqn. (10) or Fig. 23 is

λ = 0.0162 (resistance coefficient as a result of natural roughness) The sum of all loss coefficients is

514.185.0004.010003.025.1/6001.025.1/60ni =+⋅+⋅+⋅=ζ⋅Σ

The resistance coefficient λb as a result of operational roughness according to Eqn. (6) is now

λb = 0.0162 + 0.5/60 ⋅ 1.514 = 0.0288

λb applied in Eqn. (35) gives

⋅⋅⋅⋅

−⋅⋅=−

0288.05.08.01031.151.210

45.084.14k

60288.021

b

)1004844.0101317.1(855.1k 33b

−− ⋅−⋅=

mm01.2kb = for the given impounded pipeline

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b) With the conversion of the special shafts into control shafts (by raising the benching) kb improves as follows:

914.0)8Tablefrom(25.085.0514.1 C,fdni =ζ+−=ζ⋅Σ

λb = 0.0162 * 0.5/60 ⋅ 0.914 = 0.0238 kb = 0.97 mm

c) With additional channel covering the kb value again improves as follows:

05.085.0514.1ni +−=ζ⋅Σ ζ(fd,C) from Table 8) = 0.714 λb = 0.0162 + 0.5/60 ⋅ 0.714 = 0.0222 kb = 0.71 mm