LGCIE – Eaux Urbaines
Cours d’Hydrologie Urbaine
EXERCISES
Jean-Luc BERTRAND-KRAJEWSKI
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
1
CONTENTS
1. INTRODUCTION ............................................................................................................................................. 2 2. MODEL CONCEPTION.................................................................................................................................. 3 2.1 DATA AND INFORMATION............................................................................................................................... 3 2.2 QUESTIONS .................................................................................................................................................... 4 3. RAINFALL : DESIGN STORM...................................................................................................................... 5 3.1 DATA AND INFORMATION............................................................................................................................... 5 3.2 QUESTIONS .................................................................................................................................................... 6 4. RAINGAUGE CALIBRATION ...................................................................................................................... 6 4.1 DATA AND INFORMATION............................................................................................................................... 6 4.2 QUESTIONS .................................................................................................................................................... 6 5. RAINFALL MEASUREMENTS ..................................................................................................................... 7 5.1 DATA AND INFORMATION............................................................................................................................... 7 5.2 QUESTIONS .................................................................................................................................................... 7 6. RAINFALL – RUNOFF : PART 1 .................................................................................................................. 7 6.1 DATA AND INFORMATION............................................................................................................................... 7 6.2 QUESTIONS .................................................................................................................................................... 8 7. RAINFALL – RUNOFF : PART 2 .................................................................................................................. 8 7.1 MODIFICATION OF THE CATCHMENT .............................................................................................................. 8 7.2 MODIFICATION OF THE MODEL ....................................................................................................................... 8 8. INFILTRATION LOSSES ............................................................................................................................... 9 8.1 DATA AND INFORMATION ...................................................................................................................... 9 8.2 QUESTIONS .................................................................................................................................................... 9 9. FLOW ROUTING ............................................................................................................................................ 9 9.1 DATA AND INFORMATION............................................................................................................................... 9 9.2 MUSKINGUM MODEL ...................................................................................................................................... 9 9.3 OVERFLOW STRUCTURE ............................................................................................................................... 10 10. STREETER AND PHELPS (1925) MODEL UNDER STEADY CONDITIONS ................................... 10 10.1 DATA AND INFORMATION........................................................................................................................... 10 10.2 QUESTIONS ................................................................................................................................................ 11 11. SOLID TRANSPORT................................................................................................................................... 12 11.1 DATA AND INFORMATION........................................................................................................................... 12 11.2 QUESTIONS ................................................................................................................................................ 12
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
2
1. INTRODUCTION These exercises have been elaborated to show how phenomena, measurements and modelling are interdependent in urban hydrology and urban water systems. The dynamics of hydrologic, hydraulic and biochemical processes is a crucial aspect. In this context, manual “paper and pencil” resolution and modelling can not be envisaged as the amount of calculations is too important, even in simple (or simplified) case studies. Computer and software are required. In order to facilitate the work and to concentrate on fundamental aspects without having to learn any commercial modelling software which will not offer the user possibilities to handle equations and calculations, these exercises are based on Microsoft Excel. This software is not completely adequate for dynamic modelling (especially PDEs), but it is well known and does not need any long learning phase. Most of the data required for the exercises are given in different Excel spreadsheets and files, like “exercise data.xls”. All solutions can be calculated with basic Excel functions and tools, like Analysis Toolbox and Solver. As the exercises are not independent each other, most of them need the solution of some previous ones : this is clearly indicated in the text. These exercises will require personnel work from the students. They will find most of the necessary data, equations, etc. in the lecture notes. Additional information will be provided in the terms of the exercises. However, the way to solve the problem is not completely detailed and should be defined, formalised and explained by the students.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
3
2. MODEL CONCEPTION 2.1 DATA AND INFORMATION This exercise is inspired by a paper published in 1881 by Gotthilf Heinrich Ludwig HAGEN (1797-1884). Hagen, working on the uniform flow in open channels with experimental data gathered on the Gange canal between December 1874 and April 1879 by Capt. Allan Cunningham, assistant at the Thomason College of Civil Engineering in Roorkee (India), looked for relationships of the following type : Eq. 2.1
U = nh x S y
where U n h S x, y
flow velocity numerical coefficient water depth slope of the channel experimental numerical coefficients.
The above expression could be considered as prerequisite knowledge, based on previous works and analyses carried out by Hagen himself and other researchers. Indeed, since some decades, experience had strongly suggested that there was a relationship, in case of uniform flow, between the flow velocity, the slope, the water depth and the channel or pipe material. Antoine CHEZY (1718-1798) had proposed one of the first formulas in a report dated 1775, as explained more than one century later by Herschel (1897). During the 19th Century, many hydraulicians looked for widely applicable semi-empirical formulas, based on experimental data which were not always very accurately measured compared to present practice. The main objectives of the exercise are the following ones : -
learn how to calibrate some simple models, by means of both manual and computer techniques, learn how to account for parameters uncertainties in models and their links with data uncertainties, analyse how simple models could be proposed, and the influence of hypotheses on the model identification and validation, have a brief look into some historical aspects of hydraulic modelling.
Among the data sent by Cunningham, Hagen used the following 43 data series (Table 2.1), where water depth h, slope S and mean flow velocity U had been measured. In Table 2.1, the units of the variables are old non SI units : foot/second for U, foot for h and % for S. However, as this will not affect the model conception, one kept the original values and units in this exercise. As Hagen thought that Eq. 2.1 was an appropriate model in the case of uniform flow in rivers (initial hypothesis based on previous experience), his first step consisted to find the most appropriate values of n, x and y which minimised the difference between the observed and the calculated values of the velocity U. In other words, he simply calibrated the model (Eq. 2.1) by means of the least square method.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
4
series 101 103 105 113 117 119 121 124 125 127 131 132 135 136 137 138 139 151 155 158 160 162 163 173 174 175 180 181 197 201 202 204 205 212 214 215 216 217 221 222 223 224 225
mean velocity U water depth h 4.06 7.94 3.87 7.65 3.70 7.19 3.85 6.88 3.67 6.14 3.74 5.43 3.43 5.00 2.43 3.26 1.61 1.95 0.60 0.69 1.24 4.20 4.83 3.65 3.20 2.99 2.79 2.94 2.51 2.94 2.54 2.72 2.20 2.52 4.02 9.34 3.58 8.42 3.43 7.84 3.22 7.26 3.39 6.78 3.05 6.18 1.35 3.86 1.34 4.20 1.79 4.07 0.87 2.26 0.44 1.69 3.85 6.88 3.17 9.02 3.12 8.72 3.01 8.21 3.07 7.96 2.94 7.46 2.81 7.05 2.80 6.79 2.70 6.53 2.63 6.32 2.86 4.84 2.82 4.50 2.79 4.37 2.74 4.18 2.71 4.07
slope S 0.000189 0.000207 0.000222 0.000228 0.000220 0.000245 0.000240 0.000195 0.000203 0.000113 0.000025 0.000473 0.000253 0.000208 0.000200 0.000145 0.000151 0.000227 0.000217 0.000215 0.000214 0.000221 0.000171 0.000088 0.000125 0.000215 0.000148 0.000090 0.000228 0.000191 0.000200 0.000198 0.000208 0.000160 0.000146 0.000145 0.000144 0.000140 0.000295 0.000291 0.000297 0.000304 0.000306
Table 2.1 : the 43 data series provided by Cunningham (available for calculation in the Excel file “Hagen 1881 data.xls”)
2.2 QUESTIONS 2.2.1 Calibrate the model given by Eq. 2.1 using the least square method, after linearisation of the equation (logarithmic transformation, as used by Hagen with 10-logarithms).
2.2.2 Hagen found (n, x, y) = (78.325, 0.592, 0.506). Compare with the results found in question 2.2.1 and conclude about calculated results obtained with the calibrated model versus observed values.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
5
2.2.3 Evaluate the uncertainties in the parameters n, x and y. Conclude about the possibility of setting y = 1/2 = 0.5.
2.2.4 Like Hagen, one sets now y = 1/2 = 0.5. Calibrate the two remaining parameters n and x, and evaluate their uncertainties. Conclude about the possibility of setting x = 2/3 = 0.666, taking into account the fact that Hagen found (n, x) = (68.486, 0.6394).
2.2.5 Like Hagen, one sets now y = 1/2 = 0.5 and x = 2/3 = 0.666. Calibrate the remaining parameter n. Compare measured and calculated velocities.
2.2.6 Comparing calculated and measured velocities, Hagen observed that differences were far greater than uncertainties in velocity measurements. As water depth measurements were also relatively easy to make and reliable, Hagen suggested that the weakest variable was the slope S, which was more difficult to measure. In order to investigate this assumption, he calculated the slope S from other the variables : S=
-
U2 n2h4 / 3
=k
U2
Eq. 2.2
h4 / 3
Calibrate the parameter k. Compare measured and calculated slopes. Considering that the uncertainty in slope measurements could easily reach 0.0001 (according to field measurement conditions in the late 19th Century, devices used, human errors, etc.), conclude about the model quality. Suggest a final conclusion for the whole exercise.
2.2.7 References Gauckler P.G. (1867). Etudes théoriques et pratiques sur l'écoulement et le mouvement des eaux. Comptes Rendus de l'Académie des Sciences, 64, 818-822. Hagen G.H.L. (1881). Neuere Beobachtungen über die gleichförmige Bewegung des Wassers. Zeitschrift für Bauwesen, 31, 403-412. Herschel C. (1897). On the origin of the Chézy formula. Journal of the Association of Engineering Societies, 18, 363-369. Manning R. (1891). On the flow of water in open channels and pipes. Transactions of the Institution of Civil Engineers of Ireland, 20, 161-207.
3. RAINFALL : DESIGN STORM 3.1 DATA AND INFORMATION When no raingauge is available to measure local rainfall intensities, it is necessary to use synthetic or statistic rainfall data to simulate hydrologic processes. The most frequent used rainfall patterns, usually named “design storms” as they were established initially to design pipes in sewer systems, are based on statistical analysis from long time series of rainfall measurements. In the city of Lyon (France), a 14 years long data series of rainfall measurements has been used to calculate the empirical coefficients of Montana’s formula. The values are given in Table 3.1.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
6 Return period T (years) 1 2 5 10
Rainfall duration Dp (min) 6 min < Dp < 120 min 6 min < Dp < 360 min a b a b 3.9 -0.65 3.8 -0.64 4.7 -0.63 5.0 -0.66 6.2 -0.63 7.0 -0.67 8.6 -0.65 10.4 -0.71
Table 3.1 : Values of Montana’s formula coefficients a and b applicable in the area of Lyon (France)
3.2 QUESTIONS 3.2.1 Calculate the mean intensity i (in mm/h) and the rainfall depth H (mm) of a Montana type rainfall event (constant intensity) with a duration of Dp = 250 minutes and a return period T = 1 year.
3.2.2 Calculate the mean intensity i (in mm/h) and the rainfall depth H (mm) of a Montana type rainfall event (constant intensity) with a duration of Dp = 130 minutes and a return period T = 1 year.
3.2.3 Draw the corresponding hyetographs named respectively “hyeto 250” and “hyeto 130” with a time step Δt = 1 min.
3.2.4 For a catchment with a lag-time K1 = 57 min, calculate the symmetric double-triangle design storm characteristics with a return period T = 1 year for the area of Lyon (France).
3.2.5 Draw the corresponding hyetograph named “hyeto DTRI” with a time step Δt = 1 min.
4. RAINGAUGE CALIBRATION 4.1 DATA AND INFORMATION A tipping bucket raingauge is used for rainfall measurement. Before its installation on site, the raingauge has been adjusted and calibrated in the laboratory. The corresponding data are given in the spreadsheet “Raingauge calibration”. The most frequently used raingauge calibration curves are established according to the following relationship : I r = a r I m br
with
Ir Im ar, br
Eq. 4.1 actual rainfall intensity (mm/h) measured rainfall intensity (mm/h) numerical coefficients.
4.2 QUESTIONS 4.2.1 Curve fitting : according to the experimental data, calculate the values of the numerical coefficients ar and br.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
7
4.2.2 Draw the curve fitted to the experimental calibration data.
4.2.3 Determine numerically the calibration intensity Ie (in mm/h). The calibration intensity Ie is the intensity value such that Ir = Im.
5. RAINFALL MEASUREMENTS 5.1 DATA AND INFORMATION The above calibrated tipping bucket raingauge has been installed in the area of Lyon (France). The spreadsheet « Tipping dates » gives all tip dates recorded on 26/04/1995. Each tip corresponds to a water depth Ha = 0.2 mm.
5.2 QUESTIONS 5.2.1 From these data, and according to practical rules given in the lecture notes, establish the raw hyetograph, named « hyeto 1 », i.e. the hyetograph at variable time step without correction according to Eq. 4.1 as fitted in paragraph 4.2.1.
5.2.2 Calculate the hyetograph « hyeto 2 » at variable time step with correction according to Eq. 4.1.
5.2.3 Calculate the 15 first values of the hyetograph « hyeto 3 » at constant time step Δt = 1 min, with correction according to Eq. 4.1. (The complete hyetograph is given in the file “exercise data.xls”, spreadsheet “Intensity 1 min”).
5.2.4 Calculate the hyetograph « hyeto 4 » at constant time step Δt = 15 min, with correction according to Eq. 4.1. (The complete hyetograph is given in the file “exercise data.xls”, spreadsheet “Intensity 15 min”).
5.2.5 Draw all four hyetographs. Compare them and comment on the differences observed.
6. RAINFALL – RUNOFF : PART 1 6.1 DATA AND INFORMATION One assumes that the rainfall measured on 26/04/1995 (see question 5.1 above) occurred on a densely urbanised catchment whose characteristics are : Total area : A = 300 ha Impervious area : Sa = 180 ha Mean slope : s = 0.5 % Length of the main sewer branch : Lp = 3700 m The rainfall losses are estimated as follows : Initial loss : Pi = 1.2 mm Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
8 Continuous loss : Pc = 0.3 mm/h The rainfall-runoff process will be simulated by means of the linear reservoir model.
6.2 QUESTIONS 6.2.1 Calculate the value of the lag-time K1 (in minutes) with Desbordes’ formula, taking into account catchment and rainfall characteristics (for simplicity, one will assume here that rainfall duration and depth correspond to the intense duration and depth).
6.2.2 Calculate the runoff flow rates Q1(t), named “hydro 250”, “hydro 130” and “hydro DTRI”, at the outlet of the catchment, with lag-time K1 set as calculated in 6.2.1 and a time step Δt = 1 min, for the design storm hyetographs established respectively in Questions 3.2.1, 3.2.2 and 3.2.4 above.
6.2.3 Calculate the runoff flow rate Q3(t), named « hydro 3 », at the outlet of the catchment, with lag-time K1 as calculated in 6.2.1 and a time step Δt = 1 min, for the hyetograph « hyeto 3 » established in Question 5.2.3.
6.2.4 Calculate the runoff flow rate Q4(t), named « hydro 4 », at the outlet of the catchment, with a time step Δt = 1 min, for the hyetograph « hyeto 4 » established in Question 5.2.3.
6.2.5 Draw all five hydrographs, make pertinent comparison and comment.
7. RAINFALL – RUNOFF : PART 2 7.1 MODIFICATION OF THE CATCHMENT One assumes that the catchment is different, with following characteristics : Sa = 36 ha and K1 = 10 minutes. The rainfall losses Pi and Pc remain the same as in Section 6.1. Calculate and draw the five new hydrographs named « hydro 250s », « hydro 130s », « hydro DTRIs », « hydro 3s » and « hydro 4s ». Comment.
7.2 MODIFICATION OF THE MODEL One uses again the catchment characteristics and the variable values given in Section 6.1. But the linear reservoir 1 model with lag-time K1 is now replaced by a cascade of two linear reservoirs with lag-time K 2 = K1 . 2 Calculate the new hydrograph “hydro 3c” and compare it with “hydro 3”. Comment.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
9
8. INFILTRATION LOSSES 8.1 DATA AND INFORMATION Let us assume a rainfall occurring on a slightly urbanised catchment, where the rainfall losses are estimated by means of the Horton’s model. The catchment is considered as a pervious area with following characteristics : - the size characteristics remains the same as in section 6.1. - the rainfall losses are estimated by : − infiltration capacity : Fc = 10.8 mm/h. − maximal infiltration capacity at the beginning of the rainfall : F0 = 72 mm/h. − infiltration coefficient : Ki = 5.4 h-1. The Δt = 1 min time step hyetograph to be used, named “hyeto DTRI2”, is a symmetric double-triangle design storm defined by : - return period : T = 10 years. - total rainfall duration : Dp = 240 min. - total rainfall depth : H = 43.12 mm. - intense duration : 30 min. - intense depth : 30.12 mm.
8.2 QUESTIONS 8.2.1 Calculate the net rainfall intensity inets.
8.2.2 Draw the raw hyetograph « hyeto DTRI2 » and the net rainfall hyetograph « hyeto inets » with a Δt = 1 min time step.
9. FLOW ROUTING 9.1 DATA AND INFORMATION The flow Q3(t) from the catchment (i.e. hydrograph “hydro 3”) is routed in a circular trunk sewer with following characteristics : - diameter Dc = 3 m. - slope sc = 0.005 m/m. - length Lc = 5000 m. - Manning-Strickler coefficient KMS = 65 m1/3/s (mean value for good state concrete).
9.2 MUSKINGUM MODEL 9.2.1 Calculate the flow rate QMKG(t) (m3/s), named “hydro MKG”, at the outlet of the pipe by means of the Muskingum model with constant parameters K and α (use the optimised standard values as given in the lecture notes) and draw Q3 and QMKG with Δt = 1 min.
9.2.2 Instead of using α = 0.2 as in the above question, calculate now QMKG with α = 0.7. Draw the corresponding hydrograph “hydro MKG 2” and comment.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
10
9.2.3 During the storm event dated 26/04/1995, the flow rate has been measured every 5 minutes between 12:40 and 18:30 at the outlet of the Dc = 3 m trunk sewer. The values, named Qexp(t), are given in the “exercise data.xls” file, spreadsheet “Qexp”. From these values, calibrate the numerical coefficients KMKG and α of the Muskingum model in the following two cases : a) overall agreement between measured and calculated values (hydrograph “hydro MKG cal a”) b) best simulation of the flow rate peak value (hydrograph “hydro MKG cal b”).
9.3 OVERFLOW STRUCTURE One assumes now that there is an overflow structure at the outlet of the trunk sewer. This overflow is a basic horizontal crest, which height hov = 0.780 m from the pipe invert. When the flow depth is below hov, the flow is discharged in another trunk sewer for further treatment. This fraction is named Qtreat. When the flow depth is higher than hov, the fraction above the crest is discharged into a river. One names Qov(t)this fraction of the flow. All calculations will be done with QMKG (question 9.2.1) upstream the overflow structure.
9.3.1 Calculate the hydrograph « hydro Qov » and « hydro Qtreat » and draw them.
10. STREETER AND PHELPS (1925) MODEL UNDER STEADY CONDITIONS 10.1 DATA AND INFORMATION One considers a river with following characteristics : - rectangular cross section, with a bed width Briv = 4.5 m - Manning-Strickler coefficient Kriv = 32 m1/3/s - longitudinal bed slope sriv = 0.0002 m/m - flow rate (constant values) : in winter (high flow) Qriv up = 4.0 m3/s in summer (low flow) Qriv up = 1.8 m3/s - water temperature : in winter (high flow) T = 10°C in summer (low flow) T = 20°C - BOD concentration BODriv up = 5 mg O2/L. - the flow in the river is assumed to be steady and uniform (the Manning-Strickler formula may be used). A city is located along the river, with following characteristics : - number of inhabitants : 330 000 PE (people equivalent) - daily wastewater flow rate per PE : 200 L/PE/day - daily BOD load per PE : 65 g BOD5/PE/day. One assumes that all wastewater from the city is discharged at a single sewer system outlet. The model of Streeter and Phelps (1925) only considers two variables : the dissolved oxygen concentration C (mg O2/L) and the BOD concentration BOD (mg O2/L), and two elementary processes : the surface re-aeration (Eq. 10.1) and the simplified biodegradation of BOD (Eq. 10.2), which should be combined : d (C ) = K c (C s − C ) dt
Eq. 10.1
dBOD = − K L BOD dt
Eq. 10.2
with Cs the saturation concentration of dissolved oxygen. Cs varies with the temperature T (in °C) according to the following formula (Elmore and Hayes 1960), with 5 < T (°C) < 25 :
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
11
C s (T ) = 14.652 − 0.41022 T + 0.007991 T 2 − 0.000077774 T 3
Eq. 10.3
Some experiments in the river have shown that KL20 ≈ 0.2 day-1 at T = 20°C. One assumes also that KL varies with temperature according to the following Arrhenius law : K L = K L 20 θ L (T − 20)
Eq. 10.4
Experiments have shown that Arrhenius constant θL ≈ 1.047. The value of KC (day-1) is evaluated according to the following formula (Bennett and Rathbum 1972 quoted in Schuetze 1998) : K C 20 = 5.577 U riv do 0.607 hriv do −1.689 at T = 20°C
where Uriv do hriv do
Eq. 10.5
flow velocity in the river downstream the city (m/s) water depth in the river downstream the city (m).
One assumes also that KC varies with temperature according to the following Arrhenius law : K C = K C 20 θ C (T − 20)
Eq. 10.6
Experiments have shown that Arrhenius constant θC ≈ 1.024. Duration Temperature (°C) of exposure 10 °C 16 °C 20 °C 3.5 hours 1.7 (1.2) 1.9 (1.5) 2.1 (1.6) Rainbow trout 3.5 days 1.9 (1.3) 3.0 (2.4) 2.6 (2.3) 3.5 hours 0.7 (0.4) 1.1 (0.6) 1.2 (0.9) Perch 3.5 days 1.0 (0.4) 1.3 (0.9) 1.2 (1.0) 3.5 hours 0.4 (0.2) 0.6 (0.3) 1.1 (0.5) Roach 3.5 days 0.7 (0.2) 0.7 (0.7) 1.4 (1.0) Dissolved oxygen concentrations (mg O2/L) which : - permit the survival of fishes (first value) - cause a 100 % mortality (value between brackets) Species
Table 10.1 : survival DO concentrations (from Dowing and Merkens 1957)
The time step for calculation is set to Δt = 1/4 day = 6 hours.
10.2 QUESTIONS 10.2.1 One assumes that the city has no wastewater treatment plant (WWTP) and that all wastewater is discharged into the river without any treatment. For both winter and summer conditions, calculate the water depth hriv do and the flow velocity Uriv do downstream the sewer system outlet.
10.2.2 For both winter and summer conditions, calculate the dissolved oxygen concentration C(t) versus time and the dissolved oxygen concentration C(x) versus distance x from the sewer system outlet.
10.2.3 Draw the corresponding curves « C(t) winter », « C(t) summer », « C(x) winter » and « C(x) summer ».
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
12
10.2.4 Considering these curves, conclude about the survival of the species of fishes given in Table 10.1.
10.2.5 Calculate the minimum required efficiency (i.e. minimum BOD removal rate) of a future treatment plant in order to ensure the survival of trouts under summer conditions.
11. SOLID TRANSPORT 11.1 DATA AND INFORMATION One considers -
-
a circular sewer pipe : − diameter − slope − Manning-Strickler coefficient particles : d = 0.5 mm − diameter ρs = 2600 kg/m3 − density
Dc = 1.00 m sc = 0.005 m/m KMS = 65 m1/3/s
The kinematic viscosity of water at T = 12°C (typical temperature in sewer systems) is approximately ν = 1.25.10-6 m2/s.
11.2 QUESTIONS 11.2.1 Using the Shields curve, calculate the minimum water depth ycr (m) in the pipe that is necessary to initiate the motion of the deposited particles as bed load.
11.2.2 Assuming a steady and uniform flow regime in the pipe with the water depth y = 0.4 m, calcultate the bed load transport rate qB (expressed in kg/s) with the Novak-Nalluri formula.
11.2.3 With the same assumptions as above, calculate the total transport rate qT (in kg/s) with the Macke and the Ackers-White formula. Compare the results.
11.2.4 Assuming that the above flow regime is turbulent and rough, and using the appropriate formula to calculate the settling velocity w of the particles, -
evaluate if the particles can be transported as suspended load (use the van Rijn criterion) ; if yes, calculate the suspended load transport rate qS (in kg/s/m) using the Rouse and the van Rijn formulas.
Exercise terms - 17 novembre 2011 INSA de Lyon
J.L.
Bertrand-Krajewski,
LGCIE,
