Environmental Eng Assignment

15 xx 1. In the notes in Lecture 2, on Slide 6, the case of a waste flow entering a stream is considered with no fluid accumulation and no reaction (r = 0, at least over the time scale of the mixing zone). The constituent of concern is organic material containing phosphorus (P), which is a algal nutrient. The details for case we will consider here is

The concentrations are in mg P/L.

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The flow in the stream varies with season….lowest in the summer, highest with spring rains.

Prepare a plot of stream concentration after mixing (cm) (y-axis) as a function of the pre-mixing zone stream flow (x-axis), which varies from 3 to 20 m3/s. Include in your plot a horizontal line at the regulatory limit of 100 mg/L.

Comment: For this kind of problem, it might be easier for some of you to think of mixing actual volumes rather than flows (volume/time). To do that, just consider 1 s worth of each flow.

To get the mass in volume after mixing, compute

To get the volume after mixing, compute

To get the concentration after mixing, divide the first by the second.


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2. a. For a mixed batch reactor (MBR) and first order decay, if the initial concentration c. = 150 mg/L and the decay rate constant k = 0.15/day, how many days of reaction would be required to lower the concentration to 25 mg/L?



b. What would the k value need to be increased in order to get to 25 mg/L in half the time?



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3. “Half-life” is a super important concept for first order decay cases. It is the time required for co to be reduced to , in other words, when ,

a. If as in Problem 2.a, find .



b. For the k value found in Problem 2.b., find



c. We have talked in class about the concept of the “1/e time”… What are the 1e times for part a and part b, and how do they relate to the corresponding k values?




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4. For a mixed batch reactor (MBR), explain from the math why can never actually reach 0 if the decay is first order.




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5. Today is 2021-January-13. Currently the human population is 7.9 billion. The growth rate now is about 1% per year, so k = 0.01/year. If that rate constant continues to apply so that there is exponential growth,

a. in what year-month-day would we reach 10 billion?



b. how many people will be being added per day in that year?

For part a, you can just figure out how many years it will take from 7.9 billion, then with 1 y being 365 days, you can figure out the year-month-day.




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6. At constant volume, the basic no-flow MBR equation that we considered is


In the case of first order loss, we wrote r = -kc and obtained


We proceeded to integrate this, and of course obtained


One could envision adding the chemical at some mass/time rate R, without the chemical being in a flow, perhaps by tablet addition or something. The result is that the differential equation becomes


You will all be fascinated to learn (I am sure) that this is the basic approach that is used to model the concentrations of therapeutic drugs and drugs of abuse in the body. However, the concentration scale for c that is used in that case is always some mass/per kg bodyweight scale (instead of mass per volume), perhaps mg/kg. Instead of V, we then use BW = body weight so that we have



To summarize, the units we will use are

C … mg/kg

t… h

R… mg/h

BW … kg

BW is two letters, but we don’t use italics so that indicates it’s not a product of some variable B with some savable W

The k value is then the rate constant for elimination of the drug by action of all the elimination mechanisms, including breakdown in the liver and removal intact by the kidneys via urination. The overall k for elimination is the sum for those two processes, so k = ;tom + (Both rate constants act on c(t) in the same way, so we can sum them into one.)



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a. Use separation of variables and a Table of Integrals to obtain the expression for c(t) with the initial condition that c(t = 0) = 0. I.e., integrate from t = 0 to t = t to get the function c(t) as it depends on R, BW, and k for c(t = 0) = 0. Note that R/BW behaves like a constant in the integration.



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b. The antibiotic amoxicillin is said to have a half-life of about 1 h. Compute the corresponding elimination rate constant k.



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c. The usual adult dose of amoxicillin for pneumonia is 500 mg every 8 hours, so

take R = 62.5 mg/h…

for a typical adult BW,

take BW = 60 kg.

Plot c(t) vs t with t in days out to 5 days. About how long does it take to get to steady state (dc/dt = 0)?

If 1/k is a characteristic time for this problem, is the time to steady state consistent with the characteristic time?



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7. An open tank has volume of 2000 L, and an initial concentration of nitrate of 150 mg/L as N. You can’t empty the tank, all you can do is start adding water with no nitrate (cin = 0), mix the water very well as you are doing it, and let the nitrate slowly be washed out by continuous dilution. If the flow in is 10 L/min from your hose (with 10 L/min then flowing over the brim), how long is it going to take you to get the concentration down to 75 mg/L? (Is this a half-life?)



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8. For data collected between March 25 to April 4, 1992, Figure 34.B of the USGS report at https://pubs.usgs.gov/circ/circ1133/nutrients.html gives nutrient data for the Mississippi. Unlike in Figure 34.A. there is a large spike of NH4+ (which is essentially total ammonia because NH3 will be negligible at the pH of the river). It looks rather like an exponential decay of an initial spike starting at about river mile 1900. In the accompanying PowerPoint file for this HW, I give you part of this figure plus a transparent exponential function exp(-arg) with -arg varying from 0 to -4, so arg is varying from 0 to -4. At -arg = 1, c/c_o = exp(-1) = 1/e. At -arg = 2, c/c_o = exp(-2) = 1/es, and so on.

Overlay the exp function plot on the NH4+ peak to get as good a fit as you can. The x axis of the overlay where c/c_o = 0 should lie on top of the concentration = 0 line for Figure 34.B. This is where I put it to start. Note that the peak in Figure 34.B is kind of dulled, that’s surely because during the cruise they were not able to sample exactly where the peak was, so your overlay should lie above that dulled peak… From your fit,

a. pick a value of -arg and read the corresponding number of miles below the front of the spike;

b. read the corresponding value of c/c_o.

Page 9 of the report https://pubs.usgs.gov/of/1994/0523/report.pdf has incomplete river velocity data for the data collected between March 25 to April 4, 1992. The highest value of river mile for which velocity data are available is 700-954, at 5.7 km/h.



c. Lacking anything better, use that velocity to estimate the travel time t to your selected value of -arg. Remember to do any necessary conversions between miles and km.


d. Use that t to compute your estimate of the k (h-1) for decay of NH4′ in the Mississippi River during the study.



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9. Here is a variation of the problem on slide 29 of Lecture Notes 2.

The chlorine disinfectant decays with k = 0.1 day-1.


a. c in the first section at 1 km before mixing

b. c at 1 km after mixing

c. c at 4 km



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10. Extra Credit. The following article is in the Handouts module: Int. J. Environ. Sci. Technol. (2018) 15:1249-1256 “Determining ammonia nitrogen decay rate of Malaysian river water in a laboratory flume” M. Nuruzzaman, A. A. Mamun, M. N. B. Salleh. The paper gives a range for k for decay of total ammonia in tropical rivers.


a. Look at equations 1 and 2 and comment as to whether they look familiar.



b. Find the range they give for k for tropical rivers, and compare with your value for problem 8.


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