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CVEN 413 Natural Environmental Systems (Spring 2015)
Homework #4 – Transport
March 26, 2015
Due: April 2, 2015 (In class)
Note: You are allowed to discuss the problems with others but you have to work out the questions
independently. In particular, make sure that all figures and programs are your own work and not simple
copies or modifications based on other people’s work. Keep all your programs and figures for the entire
semester, in case I have questions on your solutions.
1. (40%) Yangtze River in China is the longest river in Asia and the third longest in the world. It has a
discharge rate of 30,000 m3/s.
1) If the width of the river is 1500 m, calculate and plot the depth of the river and the stream
velocity of the river for slope ranges from 0.001 to 0.005 (assuming it has a rectangular cross
section area). Explain how you solve the depth using slope=0.003 as an example. Use Manning’s
coefficient n=0.04.
2) Calculate and plot the longitudinal diffusion coefficient for pollutant transport in the river using
the following equation, again, for slope ranges from 0.001 to 0.005.
u 2 B2
E
Hu*
β = 0.41(u*/u)2
u = stream velocity (m/s),
B = stream width (m/s)
H = stream depth (m)
u* = shear velocity (m/s) u*  gHs
s = slope of the channel (unit-less)
g = gravitational acceleration constant (9.81 m/s2)
3) If we assume that the depth of the river is 5 m and does not change, but the width of the river
changes as the slope changes, calculate and plot the width of the river as a function of the slope
from 0.001 to 0.005. Explain how you solve the width using slope=0.003 as an example.
4) Calculate and plot diffusivity E for H=5 m and slope ranges from 0.001 to 0.005. Compare with
(2), do you see any differences? Explain why or why not.
5) Continue from 1, calculate and plot the diffusion time scale and the advection time scale
transport distance of 1 m to 100 km, using slope=0.003 and width=1500 m.
2. (40%) A ship collision accident leads to a rapid release of 100 tons of non-reactive contaminants in
the Yangtze River. Unfortunately, there is a drinking water treatment plant 50 km downstream. Use
the MATLAB program we provided in class, write your own script to calculate and plot the
contaminant concentration 50 km downstream of the accident from t=0 to 24 hours after the accident.
If the regulation requires the contaminant concentration be less than 10 mg/m3 in the source water,
when will it be safe to use the Yangtze River as source water again? Use cross section area and
diffusion coefficient from the previous question, using slope=0.003 and width = 1500 m. (No need to
include the MATLAB function we provided in class if you didn’t modify it). Use a grid size ∆x of 5
km for this problem. You need to determine a suitable ∆t for your calculation.
3. (20%) A waste water treatment plant continuously discharge untreated sewage, resulting in releasing
of a reactive contaminant at a mass rate of W1 to a connected three-lake system as shown below:
Q, V, C and E represents flow rate, volume, concentration and effective diffusion coefficient
(diffusion coefficient divided by distance between the lakes), respectively. Derive a set of linear
equations that describes the concentration of the pollutant in the lakes, assuming that the contaminant
undergoes a first order reaction with reaction rate coefficient of k.