Hydraulic design of partially full sewers: beyond the AS ...

1 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 The 6th Asia Pacific Water Industry Modelling Conference September 4-5, 2013, Brisbane, Australia 1. 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 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 2.

2 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.. 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 3. Sample chart ( full flow). 4. Manning chart; n= 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: Private/company spreadsheets A few dedicated commercial hydraulics software Advanced modelling software packages Sometimes unverified, usually proprietary, and often based on older textbooks 7.

3 Hydraulic foundations: steady uniform flow in circular pipes (and other pipes). Ch zy v = C R S the mother of all Ganguillet, Kutter, Bazin, empirical largely historical Agroskin, etc Gauckler- in use, recent 1 2/3. Manning* v= R S theoretical n Strickler derivation 8. Forms of the Manning equation D8/3 S 8/3. Q= D S. 8 3 2 n n 2 2 (n Q). 3/8. 3/8. n Q . 4. D= 3/8.. S S . 2. 64 2 n Q. 2/3 2 2. n Q . S= 8/3 . D. 2 16/3. D . D8/3 S. 8/3. n= D S. 8 2 Q. 3. Q. 9. partially full circular pipes 2/3. sin( ) . v p = v 1 .. 2/3. 1 sin( ) . Qp = Q 1 [ sin( )]. 2 . y . = 2 cos 1 1 2 . D . y 1 . = 1 cos . D 2 2 .. B = D sin . 2 . 10. A frequent omission in textbooks Comparison of the Manning & DW equations shows that n' is not a constant, it depends on R'. n f 6 R. Confirmed theoretical vs observed flow differences Work by Camp (Pomeroy; non-circular pipes), etc.

4 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 11. 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 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 13. Explicit equations for partly full flow Barr & Dun, Saat i, Giroud, Esen, Akgiray, etc. Often complex, range and accuracy limits Some authors consider (variable) nf: D 2/3 S 1/2. v f = ( K f ) K f . nf where Q nf n h = 1 + X sin ( X ) ; X = 1.

5 K f = 8/3 1/2 ; D S nf D. Mean error < 2% over (10o , 20o , .., 310o ,320o , ). 14. 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 f 4 Rp Rp Ap R p . = vp = v Qp = Q . fp D R A R . Similar: exponent vs 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 . 15. The Darcy*-Weisbach and Colebrook- White equations f v2 8 f Q2. S= = 2. 2 g D g D5. 1 k . = 2 log + => implicit 'f'! . f D R f . v D 4 Q. R = =. D. k . v = 2 log + 2 g D S . D D 2 g D S.. *Should be credited to D'Aubuisson de Voisin (1834) and Weisbach (1845) 16. Moody (etc.) diagrams only educational value 17.

6 The quest for explicit CW formulae (> 30). Jain 1 k . Popular = 2 log + ;J & S (1976). f D R . Barr k ~ MRE. 1 . = 2 log + . ;Barr (1977). f D R ~ MRE. Better! 1 k . Haaland = log .. + ~ MRE. f D R . Zigrang & Silvester ~ MRE. Buzelli;. Serghides Cannot fit in this cell ~ MRE. Clamond Exact! 18. 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: k sin( ) 5/8. v p = 2 log + 2 g D S 1 .. D D 2 g D S . Fact: computationally more complex =>. opportunity to make it simpler & easier 19. An improved simple' explicit CW. formula for sewerage applications Common sewer hydraulics D -> m - 2 m k -> mm mm k/D extreme values: 1 k 51 . = 2 log + . f D f R . The iteration problem is caused by the smooth'. component of f' in the CW.

7 Approximate the smooth turbulent pipe curve 20. Range on the Moody chart 21. The improved explicit f' formula 1 k . = 2 log + . f D R . MRE 1%, ARE (better than Barr/J & S). Seemingly simple but in effect involves 2 log +. 1 antilog operations R = ln(N R ). 22. A more preferable alternative formula 1 k 5 k . = 2 log log + . f D R D R . 1. = 2 log a b log ( a + b ) . f Only 2 logs => faster Sufficiently accurate: max. rel. error = in the range of interest 23. MRE table NR eps_large eps_avg eps_sm error_l error_avg error_sm +03 +04 +04 +04 +05 +05 +05 +06 +06 +06 +07 +08 +09 24. 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 solution of the Colebrook-White function.

8 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). 25. Example: rounding errors of y=x^4 for a measured x= 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= 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 26. Computational efficiency: speed & size Computers don't work like human brains, and must be handled accordingly (ln or log10? Nay, log2!)

9 Computer programming: part science, part art System/algorithm design is more crucial, microprocessor dependent improvements and tweaks have smaller effect 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 27. 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 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 28.

10 Optimisation aspects Manning formula: what 2/3. sin( ) . about a fast sin(x) with v p = v 1 .. given accuracy? 0- /2 range is sufficient 2/3. sin( ) . A basic example (~ v p = v 1 .. max. rel. error). A slightly more complex version gives ~1E-8 max. error sin( x ) x (1 + x 2 ( + x 2 )). 29. We should not stop there: computing science & art 1-sin(x)/x and/or Non-integer exponentiation Even the entire expression 2/3. Various numerical sin( ) . approaches: v p = v 1 . Polynomial functions . 2/3. sin( ) . Taylor series Chebyshev polynomials v p = v 1 . Pad approximation . Minimax rational approximation and more 30. Example: Chebyshev polynomials 2/3. sin( ) double C(const double x). 1 {. // x in [0,6]. // map x to range [-1, 1]. const double xn = x * -1;. Relative error // return Chebyshev approximation return + xn * ( + xn * ( + xn * ( + xn * ( + xn * ( + xn * ( + xn * ( + xn * ( + xn * ( ))))))))).}