CIVIL 3136 TAMU College Station, Texas (979) FAX (979) CVEN

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1 CVEN Special Topics in Mixing and Transport in the Environment Midterm Exam #2 April 22-25, 2005 Name: CIVIL 3136 TAMU College Station, Texas (979) FAX (979)

2 Instructions: This is a take-home exam. You are permitted to consult your own notes, the course notes, and the course textbook, and you may use a hand-held scientific/engineering calculator, pencils, and a ruler. You may also ask me any questions, either by (socolofs@tamu.edu) or by phone (979) Discussing the exam with anyone else or consulting any other materials will be considered as cheating. You may not search for any material on the Web, including the course web-site. Do not consult other books or look up materials in the library. Do not rely on handbooks for units conversions or other physical properties. Include a sketch and clearly state assumptions and equations used on problems requiring detailed analysis. Failure to do so will result in a lower score. Problems must be worked in the unit system in which they are specified. You may turn in additional sheets as necessary. Computer print-outs are not accepted as computers are not in the list of permitted materials. This exam is designed to require a maximum of four hours of your effort. Pace yourself accordingly. The exam is due at 1:50 p.m. on April 25, 2005, in CE 203. Problem Maximum Score Points Earned Totals: 100 Final grade: I certify by my signature below that I have not consulted any materials besides those permitted above and that the work I am submitting is my own. An Aggie does not lie, cheat or steal, or tolerate those who do. Signature: 2

3 All of the problems on this exam relate to the following design problem. A combined sewer is connected to a storage tank that can discharge to a nearby river in the case of an overflow during a storm (see figure below). CSO River Storage tank Combined sewer The design conditions for evaluating the combined sewer overflow (CSO) are for a river flow rate Q = 50 m 3 /s, effective width B = 100 m, and slope S = The Manning s roughness is taken to be n = For each of the following problems, use these conditions unless otherwise stated. 3

4 1 Empirical dispersion coefficients (25 Points) The wide-river approximation gives the hydraulic radius as R h = Bh (B + 2h) h Use this approximation to answer the following questions: (1) 1. Compute the depth h and mean velocity u using Manning s equation and the wide river approximation for the design flow. 2. Estimate the shear velocity u. 3. Compute the dispersion coefficient from the equation from Fischer et al. (1979) D L = u2 B 2 hu (2) 4. Compute the dispersion coefficient from the equation from Deng et al. (2001) with D L = 0.15 ( ) B 5/3 ( ) u 2 (3) hu 8ɛ t0 h u ( ( ) ( ) 1 u B 1.38 ɛ t0 = (4) 3520) u h 5. If Q is reduced to 30 m 3 /s, what is the new estimate for D L using the above empirical methods? Do both methods give the same trend (i.e. if one estimate increases, does the other estimate increase)? What is the relative difference in % between the two estimates? 6. Based on your calculations and experience, which formulation would you propose to use in the absence of better information and why? 4

5 2 Dye study (35 Points) To measure the dispersion coefficient more diligently, an engineering company conducted a dye study. They injected M = 1 kg of Rhodamine WT at the location of the CSO and measured the dye concentration at two downstream locations x = 86 and 172 km downstream of the injection. The measured data are provided in the following diagrams: Dye measurement at x = 86 km downstream 0.36 Concentraiton [µg/l] Time [hrs] 0.24 Dye measurement at x = 172 km downstream Concentraiton [µg/l] Time [hrs] 1. Use the data to estimate the dispersion coefficient D L. 2. If the flow rate in the river during the dye study was 30 m 3 /s with a width of 100 m, calculate the value of α in the relationship D L = αu h (5) 3. Estimate the total mass recovered in each measurement (Hint: integrate the area under the curve graphically by summing the bars in a histogram beware of units!). 4. What processes in the river might account for a loss of injected dye tracer with downstream distance? 5. Will the fact that some dye is lost affect your estimate for D L? Why or why not? Justify your answer clearly. 5

6 3 CSO Release (20 Points) During the design storm, the CSO is estimated to release stored effluent for one hour. From the CSO design, the release rate of effluent is estimated as Q 0 = 0.5 m 3 /s with a concentration of fecal coliform bacteria of #/100 ml. Answer the following questions to estimate the bacterial contamination in the stream three days after the release. 1. Can you approximate the release as an instantaneous point source? Why or why not? 2. What is the total number of bacteria released? 3. If the injection location is half-way across the stream width, at what downstream location does the bacteria cloud become well-mixed in the lateral direction? How many days after the release does the center of mass of the cloud reach this location? 4. Using a one-dimensional model, what is the maximum bacteria concentration in #/100 ml in the stream three days after the release if the bacteria are assumed not to die off? Use D L = 625 m 2 /s. 6

7 4 Bacteria die off (20 Points) Bacteria have been observed to dye off in natural streams due to natural processes such as exposure to sunlight. For the release in Problem 3, estimate the following: 1. Assuming the bacteria dye off following a 1st-order decay process with rate constant k = 2.3 day 1, what is the new estimate for the maximum bacteria concentration three days after the release in #/100 ml. 2. What is the half-life of the bacteria? 3. Where is the center of mass of the bacteria cloud when half the injected bacteria are still present in the system? 4. If the swimming standard is 2000 #/100 ml, how long after the storm will it be safe to swim? What downstream distance in the river is negatively affected with regard to the swimming standard? 7

8 A Useful relationships 1 m 3 = 1000 l 1 cm 2 = 1 ml The standard deviation of the 1-D instantaneous point source distribution is σ = 2Dt (6) If the width of an instantaneous 1-D point source distribution is measured at two different times, then the following relationship applies σ 2 2 σ 2 1 = 2D(t 2 t 1 ) (7) The two-dimensional point source solution is C(x, y, t) = M ( ) ( 4πHDt exp (x ut)2 exp y ) 4Dt 4Dt (8) The total mass injected is calculated from a concentration profile as M(t 0 ) = C(x, t 0 )Adx (9) If you are integrating data collected at a measuring station as a function of time, then: M(L) = C(L, t)audt (10) A centerline injection is well-mixed laterally in a stream at the downstream location L y when B = 2 2DL y /u (11) Manning s Equation in S.I. units is u = 1 n R2/3 h S1/2 The one-dimensional instantaneous point source solution with first-order decay is ( ) M C(x, y, t) = A 4πDt exp (x ut)2 kt 4Dt (12) (13) References Deng, Z.-Q., Singh, V. P. & Bengtsson, L. (2001), Longitudinal dispersion coefficient in straight rivers, J. Hydr. Engrg. 127(11), Fischer, H. B., List, E. G., Koh, R. C. Y., Imberger, J. & Brooks, N. H. (1979), Mixing in Inland and Coastal Waters, Academic Press, New York, NY. 8

Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118

CVEN 311-501 (Socolofsky) Fluid Dynamics Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 Name: : UIN: : Instructions: Fill in your name and UIN in the space