Hydraulic design of partially full sewers:

beyond the AS 2200-2006 Standard

Marc Couperthwaite (presenter), Laszlo Erdei and Warren Day*

School of Engineering & Surveying, University of Southern Queensland, Australia *Griffith School of Engineering, Australia

1The 6th Asia Pacific Water Industry Modelling Conference

September 4-5, 2013, Brisbane, Australia

The AS 2200-2006 Standard

• “Design charts for water supply and sewerage”, accepted in 2005, minor amendments in 2009

• Prepared by Committee PL-045:

• • Australian Chamber of Commerce and Industry

• • Australian Nuclear Science and Technology Organisation

• • CSIRO Manufacturing and Infrastructure Technology

2

• • CSIRO Manufacturing and Infrastructure Technology

• • Certification Interests (Australia)

• • Energy Networks Association

• • Engineers Australia

• • Master Plumbers, Gasfitters and Drainlayers New Zealand

• • New Zealand Water and Waste Association

• • Plastics Industry Pipe Association of Australia

• • Plastics New Zealand

• • Water Services Association of Australia

Purpose

• “The objective of this Standard is to provide

designers of pipelines for the conveyance of water

and sewerage, with a set of charts and mathematical

formulae for the determination of flow

characteristics.”

3

characteristics.”

• Provides formulae, charts for full flowing pipes

(Colebrook-White (CW) and Manning formulae), a

viscosity-temperature table, minor loss coefficient &

pipe roughness data, and a few sample calculations

Sample chart (full flow)

4

Manning chart; n=0.012

5

Partial flow chart

6

Issues

• It is suitable for occasional, manual calculations, with

a warning: “for approximate evaluations only”

• It is anachronistic in this digital age

Alternatives:

7

Alternatives:

• Private/company spreadsheets

• A few dedicated commercial hydraulics software

• Advanced modelling software packages

• Sometimes unverified, usually proprietary, and often

based on older textbooks

Hydraulic foundations: steady uniform flow

in circular pipes (and other pipes)

Chézy the mother of all

Ganguillet,

v C R S= ⋅ ⋅

8

Ganguillet,

Kutter, Bazin,

Agroskin, etcempirical largely historical

Gauckler-

Manning*–

Strickler

in use, recent

theoretical

derivation

2/31v R S

n=

Forms of the Manning equation

( )3/8

8/38/3

3

3/84

3/8

0.31169

8 2

2 21.54830

D SQ D S

nn

n Q n QD

S

π ⋅ ⋅= ⋅ ⋅

⋅ ⋅

⋅ ⋅ ⋅ ⋅ = ⋅

≃

≃

9

3/8

22/3 2 2

2 16/3 8/3

8/38/3

3

1.54830

64 210.2936

0.31169

8 2

DSS

n Q n QS

D D

D Sn D S

QQ

π

π

= ⋅

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅

⋅ ⋅= ⋅ ⋅

⋅ ⋅

≃

≃

≃

Partially full circular pipes

[ ]

2/3

2/3

sin( )1

1 sin( )1 sin( )

2

p

p

v v

Q Q

θθ

θθ θ

π θ

= ⋅ −

= ⋅ ⋅ − ⋅ − ⋅

10

1

2

2 cos 1 2

11 cos

2 2

sin2

y

D

y

D

B D

π θ

θ

θ

θ

−

⋅

= ⋅ − ⋅

= ⋅ −

= ⋅

A frequent omission in textbooks

• Comparison of the Manning & DW equations shows that ‘n’ is not a constant, it depends on ‘R’

•

• Confirmed theoretical vs observed flow differences

• Work by Camp (Pomeroy; non-circular pipes), etc.

60.23n f R≈ ⋅ ⋅

11

• Work by Camp (Pomeroy; non-circular pipes), etc. but using limited and partly questionable data

• Chart form, though several approximate (polynomial) formulae exist to calculate the ‘np/n’ factor

• The current ASCE Design Manual is based on Camp’s work

A ‘complete’ design chart

12

Using the Manning equation (also for

partly full pipes)

• Adequate for use in many applications (rough

turbulent zone)

• ‘v’ and ‘Q’ are simply obtained for full pipe flow

• Unstable region => prudent to consider

13

• Unstable region => prudent to consider

full flow = maximum design flow

• For a given relative depth ‘vp’ and ‘Qp’ are easily

obtained

• Calculations of flow depth, D, S are anything but

straightforward

Explicit equations for partly full flow

• Barr & Dun, Saatçi, Giroud, Esen, Akgiray, etc.

• Often complex, range and accuracy limits

• Some authors consider (variable) nf:2/3 1/2⋅

14

( )

( )

2/3 1/20.0036 0.336

0.360.58

8/3 1/2

o o o o o

10.18 9.61

where

; 1 0.37 sin ; 1

Mean error < over (10 , 20 , ..., 310 ,320 ,321.46 ) ±2%

f f f

f

f

f

f

D Sv K K

n

Q n n hK X X X

D S n Dπ

⋅= − ⋅ ⋅ ⋅

⋅= = + ⋅ ⋅ ⋅ = − ⋅

The European Standard

Full flow is obtained using the DW-CW equations

• More parameters, more complex but better results

The effect of variable friction factor is addressed by Franke’s elegant solution (1957), well-verified

1/8 0.625 0.6254 p p p pR R A Rf

v v Q Q⋅

= ⇒ = ⋅ ⇒ = ⋅

15

• Similar: exponent 0.625=5/8 vs 0.667=2/3 in Manning

• After some 30 years of limited use (Germany, Austria, Hungary) became the current EU standard “DIN EN 752 (2008) Drain and sewer systems outside buildings”

4 p p p p

p p

p

R R A Rfv v Q Q

f D R A R

⋅ = ⇒ = ⋅ ⇒ = ⋅

The Darcy*-Weisbach and Colebrook-

White equations

2 2

2 5

8

2

1 2.512 log => implicit 'f'!

f v f QS

g D g D

k

π⋅ ⋅ ⋅

= =⋅ ⋅ ⋅ ⋅

= − ⋅ +

16*Should be credited to D’Aubuisson de Voisin (1834) and Weisbach (1845)

2 log => implicit 'f'!3.7

4

2.512 log 2

3.7 2

R

R

D �

v D Q�

D

kv g D S

D D g

f

D

f

S

ν νπ

ν

= − ⋅ + ⋅ ⋅ ⋅ ⋅

= =⋅ ⋅

⋅= − ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

Moody (etc.) diagrams – only educational

value

17

The quest for explicit CW formulae (> 30)

Jain & Swamee;

Barr

Popular

~3.3% MRE

~3.1% MRE

Haaland

Better!

~1.4% MRE

0.9

0.89

1 5.742 log ;J & S (1976)

3.7

1 5.20162 log ;Barr (1977)

3.7

R

R

k

D �f

k

D �f

= − ⋅ + ⋅

= − ⋅ + ⋅

1.111 6.9

1.8 log3.7

k

D �f

= − ⋅ + ⋅

18

Haaland ~1.4% MRE

Zigrang & Silvester

Buzelli;

Serghides

Clamond

Cannot fit in this cell

~0.13% MRE

~0.024% MRE

Exact!

1.8 log3.7 RD �f

= − ⋅ + ⋅

Using the DW-CW equations for

partly full pipes

• Only ‘v’ and ‘Q’ can be obtained relatively

simply, partially full pipes give extra

complication. The easiest partial flow case:

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• Fact: computationally more complex =>

opportunity to make it simpler & easier

5/82.51 sin( )

2 log 2 13.7 2

p

kv g D S

D D g D S

ν θθ

⋅ = − ⋅ + ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅

An improved ‘simple’ explicit CW

formula for sewerage applications

• Common sewer hydraulics

• D -> 0.1 m - 2 m

• k -> 0.5 mm – 1.5 mm

• k/D extreme values: 21 .51k

20

• 0.005 – 0.00075

• The iteration problem is caused by the ‘smooth’ component of ‘f’ in the CW

• Approximate the smooth turbulent pipe curve

21 .2 log

3.7

51

R

k

D ff �

= − ⋅ + ⋅ ⋅

Range on the Moody chart

21

The improved explicit ‘f’ formula

• MRE 1%, ARE 0.3% (better than Barr/J & S)

• Seemingly simple but in effect involves 2 log +

0.9

1 5.382 log

3.7 R

k

D �f

= − ⋅ + ⋅

22

• Seemingly simple but in effect involves 2 log +

1 antilog operations

0.9 (N )0 n9 l. R

R� e ⋅=

A more preferable alternative formula

( )12 log log 2.

1 5 13.52 log log

3.7

7

3.7 R R

k k

D � Df

b

�

a a bf

= − ⋅ − + ⋅ ⋅

= − ⋅ − ⋅ + ⋅

23

• Only 2 logs => faster

• Sufficiently accurate: max. rel. error = 0.2% in the range of interest

( )2 log log 2.7ba a bf= − ⋅ − ⋅ + ⋅

MRE table

NR eps_large eps_avg eps_sm error_l error_avg error_sm

4.E+03 0.005 0.0029 0.00075 0.52% 0.55% 0.57%

1.E+04 0.005 0.0029 0.00075 0.12% 0.09% -0.03%

2.E+04 0.005 0.0029 0.00075 0.04% 0.01% -0.20%

5.E+04 0.005 0.0029 0.00075 0.02% 0.00% -0.19%

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1.E+05 0.005 0.0029 0.00075 0.01% 0.01% -0.11%

2.E+05 0.005 0.0029 0.00075 0.01% 0.01% -0.04%

5.E+05 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

1.E+06 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

2.E+06 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

5.E+06 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

1.E+07 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

1.E+08 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

1.E+09 0.005 0.0029 0.00075 0.00% 0.00% 0.00%

Accuracy of calculations

• Primarily it is a computational & modelling issue

• But not only of academic interest

• “Practical considerations often mitigate against there

being great significance in the accuracy of the

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being great significance in the accuracy of the

solution of the Colebrook-White function. However,

it is good practice to achieve accurate solutions and

then apply engineering judgement rather than to

obtain approximate solutions and then apply larger

safety factors.“ (Prof. Barr)

Example: rounding errors of y=x^4 for

a measured x=1.9634954

26

•Problem specific: astronomers may have calculate with >100 figures to avoid

gross errors. If unsure, check it! DW formula involves similar exponent in D

•In hydraulics 3 significant digits (in results) often suffice; given g=9.81m/s2

•The CW formula itself is not highly accurate but its sufficiently accurate

resolution is needed for numerical simulations (need for repeatability)

•One should prefer accurate AND fast calculations, either on old or new PCs

Computational efficiency: speed & size

• Computers don’t work like human brains, and must be handled accordingly (ln or log10? Nay, log2!)

• Computer programming: part science, part art

• System/algorithm design is more crucial, microprocessor dependent improvements and tweaks have smaller effect

27

• Direct translation/coding of ‘meaningful’ (for us) formulae is typically inefficient (but kudos for allowing maintainability and portability)

• Clarity vs speed vs accuracy => compromise

• Buggy bloatware is spreading, and we are often forced to use such

Fast and slow math operations

• DIFFICULT to get reliable metric and measure ‘speed’

• Very fast: +, -, abs, register shifts; etc. Clock cycles (c) = 1

• Fast: * (c3), / (c3-c10)

• Typically slow: transcendental functions (c over 100)

• Square root, ln, log, exp, non-integer power

• Trigonometric functions

28

• Trigonometric functions

• Integer libraries vs floating point numbers

• Math co-processor: practically granted on PCs

• Look up tables (LUT) => loosing/lost advantage; memory operations are relatively slow on current hardware

• CUDA & parallel systems: for certain problems

• Compiler optimisation vs assembler coding

Optimisation aspects

• Manning formula: what

about a fast sin(x) with

given accuracy?

• 0-π/2 range is sufficient

2/3sin( )

1pv vθθ

= ⋅ −

29

• A basic example (~0.2%

max. rel. error)

• A slightly more complex

version gives ~1E-8 max.

error

2/3sin( )

1pv vθθ

= ⋅ −

2 2(1 ( 0.1660sin( ) 5 0.00761 ))xx x x⋅ + − + ⋅≅ ⋅

We should not stop there: computing

science & art

• “1-sin(x)/x” and/or

• Non-integer exponentiation

• Even the entire expression

• Various numerical approaches:

/2 3sin( )

1pv vθθ

= ⋅ −

30

approaches:

– Polynomial functions

– Taylor series

– Chebyshev polynomials

– Padé approximation

– Minimax rational approximation

– … and more

2 3/sin( )

1

p

pv vθθ

θ

−

= ⋅

Example: Chebyshev polynomials

Relative error

0.40%

• double C(const double x)

• {

• // x in [0,6]

// map x to range [-1, 1]

const double xn = x * 0.3333333333333333 -1;

// return Chebyshev approximation

• return 0.9684736662906951

• + xn * ( 0.7024722360781829

• + xn * ( -0.6907372065827905

2/3sin( )

1θθ

−

0.00%

0.20%

0.3 1.2 2.1 3 3.9 4.8 5.7

theta

• + xn * ( -0.6907372065827905

• + xn * ( -0.1786818406654233

• + xn * ( 0.26499779029867476

• + xn * ( -0.01366365377447798

• + xn * ( -0.08515204660857621

• + xn * ( 0.05565480177863434

• + xn * ( 0.0568958604526894

• + xn * ( -0.04949435049818298)))))))));

• __________________________________________

• 0.27% max. error @ 0.48 radian:

• < 0.08 mm water depth difference for a 2.0 m diameter pipe

[3,3] minimax approximation

2 32/3

2/3

2 3

sin( )1

sin( )1 ; Horner form

( 1 2 3 4 )

( 5 6 7 8 )

1 ( 2 ( 3 4 ) )

5 ( 6 ( 7 ) )8

c c x c x c x

c c x c x c x

c c c c x x x

c c c c x x x

θθ

θθ

+ − =

+ ⋅ + ⋅ ⋅+ ⋅ + ⋅ + ⋅

+ + + ⋅ ⋅− =

⋅+ + + ⋅ ⋅ ⋅

32

1 1.096075E-4; 2 5.3490661E-3; 3 9.874888233E-2; 4 5.442214c c c c= = = = − 664E-3

5 9.42256151E-2; 6 2.815307648E-1; 7 5.041846663E-2; 8 1.099009467E-2

Fits in an Excel cell even without using a table for the constants (x = cell reference)

=(0.1096075e-3+(0.53490661e-2+(0.

c c c c= = = − =

9874888233e-1-0.5442214664e-2*x)*x)*x)/

(0.942256151e-1+(.2815307648+(-0.5041846663e-1+0.1099009467e-1*x)*x)*x)

Adequate accuracy

• Max. error [0..6] (y/D=0.94) < 0.12%

33

Continued-fraction form (RHS only)

• 6 additions + 3 divisions versus

• 6 additions + 1 division + 6 multiplications

• Usually worth consideration/test

•

34

•

Proof of concept: an XLL add-on

• In the making

• Main purpose: testing & verifications

• Excel as user interface

• Coding in C/C++, with some parts improved in

35

• Coding in C/C++, with some parts improved in

x86 assembler

• Orders of magnitude faster than using xla, vba

macros etc.

Summary

• Shortcomings of the AS 2200-2006 Standard

• Hydraulics of full and partially full pipes

• Using the Manning formula, variable ‘n’

• Explicit equations for partly full flow• Explicit equations for partly full flow

• The continental approach (DW & CW formulae)

• Two explicit ‘range’ formulae for CW

• Efficient computation of the ‘Rp’ (and Franke’s

factor)