1d Model Civil and Surveying Sotware Version 7 Drainage Analysis Module Hydraulics Owen Thornton BE (Mech), 1d Model Programmer owen.thornton@1d.com 9 December 005 Revised: 10 January 006 8 February 007 This document describes the various terms relevant to the hydraulic calculations perormed by the Drainage Analysis module o 1d Model version 7, including how and where they are calculated. The terms pipe and channel are sometimes used interchangeably, as are the terms pit, manhole, inlet and structure. Note: text in red is only applicable to version 7.0C1g and beyond. Page 1 o 10
Q = Peak Flow The peak low rate in the pipe, determined rom the contributing catchments via the Rational Method (and including any direct pit or pipe low that may have been speciied upstream). It is shown in the Hydraulic Report as "Pipe Flow", and is also set as a pipe attribute named "pipe low". V = Q / A Velocity The velocity in the pipe when the peak low, Q, ills the entire cross-sectional area, A, o the pipe. This is the minimum velocity possible or this low rate. It is shown in the Hydraulic Report as "Full Pipe Velocity", and is also set as a pipe attribute named "ull pipe velocity". Note: V is the only velocity ever used to determine pressure head changes through the upstream pit, because the pressure head change actors, K u, suggested in all the dierent publications I have ound, are only applicable to pressurised pipes. Q cap = Capacity Flow For the particular size, grade and roughness o the pipe, and assuming no downstream tailwater restrictions, the capacity low is the low rate theoretically possible in the pipe, at the point where the low would become pressurised (i.e. lowing exactly ull). It is shown in the Hydraulic Report as "Capacity Flow", and is also set as a pipe attribute named "low capacity". Note: In all versions prior to 7.0C1g, the value or the capacity low is actually Q mcap (see below). The change has been made or version 7.0C1g and beyond, because the new Q cap values are a better match to the capacities published by pipe manuacturers, and also because o the inherent instability o the unpressurised low, Q mcap, which can become pressurised by only the slightest increase in rictional resistance beyond the design igure. V cap = Capacity Velocity The velocity corresponding to Q cap. That is: V cap = Q cap / A. It is set as a pipe attribute named "capacity velocity". Capacity Ratio = Q / Q cap This is simply the ratio o the peak low to the capacity low. I greater than 1.0, Q will always be pressurised. I less than 1.0, Q may or may not be pressurised, depending on the conditions downstream. Some governing bodies speciy that it should always be less than 1.0. It is set as a pipe attribute named "low capacity ratio". Q mcap = Max Capacity Flow Similar to Q cap, but representing the maximum unpressurised low rate theoretically possible in the pipe. For a circular pipe, Q mcap occurs at a depth o about 0.94xD. For a box-culvert, Q mcap occurs at a depth just below the obvert (0.999xH, say). For circular pipes, Q mcap is approximately 1.07xQ cap. For box-culverts, Q mcap can be considerably greater than Q cap, due to the sudden increase in riction losses when the low comes into contact with the top wall an eect more pronounced as the culvert's width/height ratio increases. It is set as a pipe attribute named "low capacity max". V mcap = Max Capacity Velocity The velocity corresponding to Q mcap. At whatever depth Q mcap occurs, the wetted cross-sectional area, A w, is calculated so that: V mcap = Q mcap / A w. It is set as a pipe attribute named "capacity max velocity". Capacity Max Ratio = Q / Q mcap This is simply the ratio o the peak low to the max capacity low. It is set as a pipe attribute named "low capacity max ratio". Page o 10
d n = Normal Depth The depth in the pipe when the slope o the water surace o the peak low, Q, is parallel to the pipe slope. I Q is greater than Q cap, d n is set to the obvert o the pipe. It is set as a pipe attribute named "normal depth", and the value [d n / <pipe height>] is set as "normal depth relative". d c = Critical Depth The depth in the pipe when the local energy head (the sum o the pressure and velocity heads only, ignoring the gravity head) o the peak low, Q, is at a minimum. It is set as a pipe attribute named "critical depth", and the value [d c / <pipe height>] is set as "critical depth relative". V n = Normal Depth Velocity The velocity in the pipe when the peak low, Q, is lowing at the normal depth, d n. This is the maximum velocity possible or this low rate, in the most common case o a "steep" slope pipe (where d c > d n ). In the rare case o a "mild" slope pipe (where d c < d n ), it is possible or the velocity to be slightly greater near the downstream end o the pipe, i the low "drops o the end" and the depth approaches d c. It is shown in the Hydraulic Report as "N Depth Velocity", and is also set as a pipe attribute named "normal velocity". Note: V n is the velocity against which V min and V max are tested. V c = Critical Depth Velocity The velocity in the pipe when the peak low, Q, is lowing at the critical depth, d c. When the low is slower than V c (deeper than d c ) it is termed tranquil low. When the low is aster than V c (shallower than d c ), it is termed rapid low. It is set as a pipe attribute named "critical velocity". V min & V max = Allowable Velocity Limits These velocity limits are oten speciied by governing bodies, to ensure that the low is ast enough to wash away any impediments, but not so ast as to scour the pipes. They can be set on the "GLOBAL->Main" tab o the Drainage Network Editor, and are checked to ensure that V n lies between the two values. V a = Actual Velocity (VicRoads manual) In the Victorian "VicRoads" manual, Figure 7.4.7 shows a chart or determining the so-called Actual Velocity, V a, in part-ull pipes. The chart shows a relationship between Q / Q cap values and V a / V cap values. I have determined that the resultant V a values rom this chart, match very closely with the normal depth velocity, V n, values calculated by 1d Model. Page 3 o 10
Manning's Formula versus the Colebrook-White Formula The Drainage Analysis module allows pipe roughness to be speciied as either Manning's n (based on metre-second units), or Colebrook's k (speciied in mm in 1d). The choice o roughness type governs which o the two empirical ormulae is used to calculate: Q cap, V cap, Q mcap, V mcap, d n and V n. It also governs how the riction slope is determined in both the Darcy-Weisbach equation and the Gradually Varied Flow equation, when calculating the HGL values along the pipe. The Colebrook- White ormula may be seen as an even combination o the Rough and Smooth Pipe Laws o turbulent low, which were both derived rom boundary-layer theory by Prandtl and von Kármán, but adjusted to match the experimental results o Nikuradse, who perormed a series o lab tests on small pipes o uniorm roughness. Colebrook and White, however, perormed tests on a range o commercial pipes (o non-uniorm roughness) with an equivalent k deduced rom the constant riction actor,, observed at high Reynolds number and ound that or most o the pipes they tested, their new ormula matched well. The turbulent range o the now widely-used Moody Diagram, is wholly based on the Colebrook-White ormula, and represents the trend to be expected, in the absence o any speciic data or particular pipes. Where non-circular pipes are used, or where part-ull low occurs in any kind o pipe or open channel o reasonably regular cross-section, substitution o 4R or D (our times the hydraulic radius or diameter) can be justiied, with the assumption that the mean shear-stress around the wetted perimeter is not too dissimilar rom the uniorm stress around a circular perimeter (true enough or box-culverts and most regular open channels). Because the Colebrook-White ormula accounts or the variations in dependent on the relative roughness (k/r or k/d) and the Reynolds number, it is slightly more reliable, when considered across the range o pipe sizes and low rates commonly ound in stormwater systems. Most o the values published or Manning s n are or large open channels, where the dependency on Reynolds number is slight, and the relative roughness is implied in the n value. As such, Manning s ormula is good or larger systems (especially natural channels), but when applied to smaller systems like typical stormwater pipes, the Colebrook-White ormula suggests that these published n values should be reduced somewhat. Overall, the primary diiculty with both ormulae lies in the selection o suitable n or k values, and signiicant errors are not uncommon. The ollowing igure shows a normalised graph o Manning's ormula, or a ull range o low depths in both circular pipes and (square) box-culverts. (The corresponding Colebrook-White graph is so similar at this scale that it may be considered identical.) Note the dierence between Q cap and Q mcap on the graph. Page 4 o 10
The Hydraulic Grade Line (HGL) In 1d Model, when a Drainage string is proiled in a Section View, or plotted with a Drainage Longsection PPF, you have the option to display the HGL that results rom a storm analysis o the network to which the string belongs. The HGL represents the peak sum o the pressure and gravity heads at every point in the network, and in 1d, it is idealised to show the separate head losses that occur along the pipes and through the pits. HGL Along Pipes: The HGL drawn along a pipe in 1d, is simply the straight line joining the pipe s entrance and exit HGL levels. As such, in pipes where the steady low is wholly pressurised, it is a true representation o the riction slope, and consequent loss o total head due to riction. However, where part-ull low occurs in any portion o the pipe, neither the riction slope nor the water surace slope is constant, and so the straight line is not a true HGL, and does not imply anything other than the true entrance and exit HGL levels. In these cases o part-ull low, the true HGL is calculated internally by numerical integration o the Gradually Varied Flow equation, handling all possible cases o tranquil and rapid low on mild and steep slope pipes, including the hydraulic jumps that can occur on steep slopes. The true HGL or part-ull low may be shown in uture versions o 1d, especially i drawing standards start to dictate it. But, due to the non-uniorm velocity head, the loss o total head due to riction cannot be determined rom this true HGL, and must instead be determined rom the Total Energy Line (TEL). HGL At Pits: The horizontal HGL drawn across a pit, represents the peak water surace elevation (WSE) in that pit, and is tested against the reeboard and surcharge limits imposed there. Also, i this WSE is higher than the level ormed by the minimum o d n and d c at the exit o any pipe lowing into the pit, then the WSE orms a tailwater condition or those upstream pipes. The dierence between the WSE and the HGL at the entrance o the downstream pipe, represents the change (normally a loss) in pressure head through the pit (it does not represent the total head loss through the pit). Schematic o a typical HGL in a steep slope pipe Page 5 o 10
Hydraulic Equations: Bernoulli s Equation (or the head o a streamline) V HT = HV + H P + HG = + h + g z Reynolds Number (laminar low < 000 < critical zone < 4000 < turbulent low) V D 4 V R Re = = ν ν Darcy-Weisbach Equation (or steady, uniorm [pressurised] low) H T V V m V = S = = = L D g 4R g R g Gradually Varied Flow Equation (or steady, non-uniorm [open channel] low) dh S0 S = dl V B 1 g Aw Manning's Formula (or complete turbulence, high Re) V 8 1 6 g g R = = (Manning s relation to Chézy s coeicient, or metre-second units) m n 1 3 S R = (or metre-second units) OR n V 1 3 S R = 1.49 (or oot-second units) n Colebrook-White Formula (or transition zone low and complete turbulence, Re > 4000) 1 k.51 = log + 10 (as published by Colebrook, 1939) 3.7 D Re V = k.51ν = k gds log10 + 4 grs log10 + 3.7 D D gds 14.8 R Pressure Head Change Formula (through pit directly upstream o pipe) H = K V / g P u 0.314ν R grs Page 6 o 10
where: [dimensions] H T = total head (energy per unit weight o water) [L] H V = velocity head = V / g [L] H P = pressure head = h [L] H G = gravity head = z [L] V = mean velocity o low through a cross-section [L/T] g = acceleration due to gravity [L/T ] h = pressure head (and depth o low in an open channel) at a cross-section [L] z = height o channel invert at a cross-section, rom a constant horizontal datum [L] Re = Reynolds number [-] ν = kinematic viscosity o water [L /T] D = diameter o circular pipe [L] R = hydraulic radius o low at a cross-section = A w /P w = D/4 or ull circular pipe [L] A w = wetted cross-sectional area [L ] P w = wetted cross-sectional perimeter [L] L = plan length o channel [L] = Darcy riction actor (based on D) [-] m = Fanning riction actor (based on R) = /4 conusingly, oten denoted by [-] dh/dl = longitudinal slope o water surace at a cross-section, relative to S 0 [-] B = lateral breadth o water surace at a cross-section [L] S 0 = channel slope [-] S = riction slope (energy lost, per unit weight o water, per unit length o channel) [-] S = context dependent typically: S 0 when solving or V; S when solving or S [-] n = Manning roughness actor [T/L 1/3 ] k = Colebrook roughness [L] K u = pressure head change actor or pit/manhole/structure directly upstream o pipe [-] V = mean velocity o pressurised low (ull low) at pipe entrance [L/T] Page 7 o 10
Determining the Critical and Normal Depths in Part-ull Pipes Wherever the peak low, Q, is less than the capacity low, Q cap, the potential exists or part-ull (open channel) low to occur. In these instances, the critical and normal low depths are determined by 1d Model, using the equations shown in the igure below. Note that the graph within the igure, shows the solutions or a circular pipe on a normalised scale. Page 8 o 10
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More Hydraulic Equations (or reerence only not used in 1d): Chézy s Formula (or complete turbulence, high Re) 8gRS V = = C R S (C is Chézy s coeicient) Laminar Flow Law (or Re < 000) 64 = (via Poiseuille s ormula) Re Smooth Pipe Law (or Re > 4000, very low k/d) 1.51 = log 10 (adjusted to Nikuradse s data rom uniorm roughness pipes) Re 7 1.8 log10 (Colebrook s approximation: within ±1.5% or 5000 < Re < 10 8 ) Re 5.76 log10 0. 9 Re Rough Pipe Law (or complete turbulence, high Re) 1 k = log10 (adjusted to Nikuradse s data rom uniorm roughness pipes) 3. 7 D 170 D Re > (range o Rough Pipe Law, based on Nikuradse s observations) k 1650D Re > (range o Rough Pipe Law, based on C-W ormula, or accurate to < 1%) k Converting k to n (by equating Manning s ormula to the Colebrook-White ormula) n = R 1 6 k 4 g log10 + 14.8 R 0.314ν R grs Sources: Colebrook, C.F., 1939, Turbulent Flow in Pipes with particular reerence to the Transition Region between the Smooth and Rough Pipe Laws, J. Instn. Civ. Engrs., Vol. 11, 133-156. Massey, B.S., 1989, Mechanics o Fluids, 6 th Ed., Chapman & Hall. Pilgrim, D.H. (Ed.), 1987, Australian Rainall and Runo, Vol. 1, Instn. Engrs. Aust. Page 10 o 10