1d Model Civil and Surveying Sotware Drainage Analysis Module Hydraulics Owen Thornton BE (Mech), 1d Model Programmer owen.thornton@1d.com 04 June 007 Revised: 3 August 007 (V8C1i) 04 February 008 (V8C1p) This document deines the various terms relevant to the hydraulic calculations perormed by the Drainage Analysis module o 1d Model, 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. Page 1 o 11
1d Model Drainage Analysis Module Hydraulics Deinitions: Q rat = Peak Flow The peak low rate or the pipe, without consideration o bypass lows. It is determined rom the contributing catchments via the Rational Method (and includes any direct pit or pipe low that may have been speciied upstream). It is shown in the Hydraulic Report as "Peak Flow Qrat", and is also set as a pipe attribute named "calculated peak low". Q b = Net Bypass Flow When considering bypass lows, Q b or the pipe is determined rom an analysis o all the upstream pits contributing to the pipe, as recommended in the Australian Rainall and Runo. It represents the sum o the peak bypass lows approaching these pits, minus the sum o the peak bypass lows leaving them. It is shown in the Hydraulic Report as "Net Bypass Flow Qb", and is also set as a pipe attribute named "calculated net bypass low". Note: the peak bypass low leaving each pit, is determined rom the pit inlet capacity data, deined or each pit type in the drainage.4d setup ile, as well as the relevant choke actor determined or each pit. Q = Pipe Flow = Q rat + Q b The peak low rate in the pipe, represented by the sum o Q rat and Q b. It is shown in the Hydraulic Report as "Pipe Flow Q", and is also set as a pipe attribute named "pipe low". Note: it is only when considering bypass lows, that Q may dier rom Q rat. Q x = Excess Pipe Flow I the pipe low, Q, is so large that the HGL would rise above the upstream pit s grate level, then Q x will represent the dierence between Q and the low rate that would keep the HGL just at the grate level. It is shown in the Hydraulic Report as "Excess Pipe Flow Qx", and is also set as a pipe attribute named "calculated excess low". Note: the excess low amount, Q x, is still included in the pipe low, Q, and is not automatically re-routed to the overland system. However, when considering bypass lows, the user may set the Qx routing increment to a value greater than zero, which will initiate extra analysis passes to re-route just enough low as is necessary, rom the pipe network to the overland system, so as to eradicate the excess pipe lows. At each extra pass, an amount no greater than the Qx routing increment is removed rom the pipe network rom the most upstream pipe(s) with excess low(s) and re-routed to the overland system. I low is entering rom the top o a pipe s upstream pit, the re-routing is achieved initially by reducing the pit s choke actor by an amount calculated to have the same eect. Once the choke actor is reduced to zero, however, any remaining Q x is re-routed as pit surcharge low, Q s. Q s = Pit Surcharge Flow When (and only when) excess pipe lows are re-routed to the overland system, it is possible or some pits to surcharge (low rising up and exiting the top o the pit). At such pits, the choke actor will have been reduced to zero, and Q s will have a positive value equal in magnitude to the negative pit inlow. In the Hydrology Report, a negative value o "Inlet Flow Qi" indicates a surcharge low, and Q s is also set as a pit attribute named "calculated surcharge low". Note: pit surcharge low is automatically included in the bypass low leaving a pit, and may re-enter the pipe network elsewhere. Page o 11
1d Model Drainage Analysis Module Hydraulics Deinitions: cont V = Full Pipe Velocity = Q / A The velocity in the pipe when the pipe 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 Vel V=Q/A", and is also set as a pipe attribute named "ull pipe velocity". Note: V is the velocity used to determine pressure head and water surace elevation losses through the upstream pit. Typically, the loss coeicients K u & K w only apply to pipes under pressure. 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 Qcap", and is also set as a pipe attribute named "low capacity". V cap = Capacity Velocity = Q cap / A The velocity corresponding to Q cap. It is shown in the Hydraulic Report as "Capacity Vel Vcap=Qcap/A", and is also set as a pipe attribute named "capacity velocity". Note: V cap is sometimes reerred to by others as at grade velocity, or more conusingly, as ull pipe velocity. Great care should be taken, to avoid conusing V cap with V. Capacity Ratio = Q / Q cap This is simply the ratio o the pipe 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 authorities speciy that it should always be less than 1.0. It is shown in the Hydraulic Report as "Q/Qcap Ratio", and is also 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 may 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. Note: Q mcap is not calculated by 1d Model, as it is an inherently unstable low, which can become pressurised with only the slightest increase in rictional resistance. Pipe manuacturers typically publish capacities which match more closely to Q cap. V mcap = Max Capacity Velocity The velocity corresponding to Q mcap, that is: V mcap = Q mcap / A w, where A w is the wetted crosssectional area at the depth at which Q mcap occurs. Page 3 o 11
1d Model Drainage Analysis Module Hydraulics Deinitions: cont d n = Normal Depth The depth in the pipe when the slope o the water surace o the pipe 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 pipe 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 pipe 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 "Norm Depth Vel Vn=Q/An", and is also set as a pipe attribute named "normal velocity". V c = Critical Depth Velocity The velocity in the pipe when the pipe 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 shown in the Hydraulic Report as "Crit Depth Vel Vc=Q/Ac", and is also set as a pipe attribute named "critical velocity". V min & V max = Allowable Velocity Ranges or V n and V cap These velocity limits are oten speciied by governing authorities, to ensure that all pipe lows are ast enough to wash away impediments, but not so ast as to scour the pipes. I set on the GLOBAL=>Main tab o the Drainage Network Editor, they are checked against the calculated V n and/or V cap values. I ever a calculated velocity is too low or too high, a new pipe grade is recommended in the Output Window, to help remedy the problem. V a = Actual Velocity (VicRoads manual) The Victorian "VicRoads" manual reers to 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. The resultant V a values rom this chart, are equivalent to the normal depth velocity, V n, values calculated by 1d Model. Page 4 o 11
Manning's Formula versus the Colebrook-White Formula: 1d Model Drainage Analysis Module Hydraulics 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 determines the ormula Manning or Colebrook-White used to calculate: Q cap, V cap, 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 empirical 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 combination-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 uniorm crosssection, 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 many 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 (Re), 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 relatively large open channels, where the dependency on Re is slight, and 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 on this normalised scale, that it may be considered identical.) Note the dierence between Q cap and Q mcap on the graph. Page 5 o 11
1d Model Drainage Analysis Module Hydraulics The Hydraulic Grade Line (HGL): In 1d Model, when a Drainage string is proiled or plotted, you have the option to display the HGL determined rom a storm analysis o the drainage network. The HGL represents the peak sum o the pressure and gravity heads throughout the network, and in 1d, it is idealised to show separate head losses along the pipes and through the pits. In reality, these losses are not separate, but continuous, with the oten highly un-developed low near the pits, progressively orming ully-developed, onedimensional low, in the pipe reaches suiciently distant rom the pits. HGL Along Pipes: The HGL drawn along a pipe in 1d, is simply the straight line joining the pipe s idealised entrance and exit HGL levels. As such, in pipes where the steady low is wholly pressurised, it is a good representation o the riction slope, and consequent loss o total head ( H T ) 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 implies nothing other than the idealised entrance and exit HGL levels. For these cases, the idealised HGL is calculated internally, assuming ully-developed low, via numerical integration o the Gradually Varied Flow equation. This handles all possible cases o tranquil and rapid low on mild and steep slope pipes, including the hydraulic jumps that can occur on steep slopes. Note that, due to the non-uniorm velocity head in these cases, H T due to riction cannot be determined rom the idealised 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. The jump in the HGL between a pit s entrance and exit, represents the change (normally a loss) in pressure head ( H P ) through that pit (it does not represent the loss in total head). I a pit s entrance HGL level is higher than the level ormed by the minimum o d n and d c in an upstream pipe, then this HGL level also orms a tailwater condition or that upstream pipe. Note that WSE is typically equal to, or slightly greater than H P, and that both these jumps may, to a certain extent, be thought o as the eect that a pit has on what would otherwise be ully-developed, one-dimensional low. Schematic o a typical HGL in a steep slope pipe Page 6 o 11
1d Model Drainage Analysis Module Hydraulics 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 [ree-surace] 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 WSE Change Formula (between pit directly upstream o pipe and pipe entrance) WSE = K V / g w Page 7 o 11
1d Model Drainage Analysis Module Hydraulics 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 actor [L] WSE = water surace elevation in pit [L] K u = pressure head change actor or pit directly upstream o pipe [-] K w = WSE change actor or pit directly upstream o pipe [-] V = ull pipe velocity [L/T] Page 8 o 11
1d Model Drainage Analysis Module Hydraulics Determining the Critical and Normal Depths in Part-ull Pipes: Wherever the pipe low, Q, is less than the capacity low, Q cap, the potential exists or part-ull (ree surace) 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 9 o 11
Page 10 o 11 1d Model Drainage Analysis Module Hydraulics
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 1d Model Drainage Analysis Module Hydraulics 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 11 o 11